On Computing the Distinguishing Numbers of Planar Graphs and Beyond: a Counting Approach

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1 On Computing the Distinguishing Numbers of Plnr Grphs nd Beyond: Counting Approch V. Arvind The Institute of Mthemticl Sciences Chenni , Indi Christine T. Cheng Deprtment of Electricl Engineering nd Computer Science University of Wisconsin-Milwukee Milwukee, WI Nikhil R. Devnur College of Computing Georgi Institute of Technology Atlnt, GA Februry 21, 2008 Abstrct A vertex k-lbeling of grph G is distinguishing if the only utomorphism tht preserves the lbels of G is the identity mp. The distinguishing number of G, D(G), is the smllest integer k for which G hs distinguishing k-lbeling. In this pper, we pply the principle of inclusion-exclusion nd develop recursive formuls to count the number of inequivlent distinguishing k-lbelings of grph. Along the wy, we prove tht the distinguishing number of plnr grph cn be computed in time polynomil in the size of the grph. 1 Introduction A vertex k-lbeling of grph G is mpping φ : V (G) {1, 2,..., k}. It is sid to be distinguishing if the only utomorphism tht preserves the lbels of G is the identity mp. The distinguishing number of G, D(G), is the minimum number of lbels needed so tht G hs distinguishing lbeling. The notion of distinguishing numbers for grphs ws first introduced nd developed by Albertson nd Collins [3]. Their focus ws on determining the reltionships between grph s utomorphism group nd its distinguishing number. Their work hs since been extended in mny directions by reserchers for grphs nd groups (e.g., [2, 5, 8, 6, 7, 10, 12, 16, 17, 20, 21]). A preliminry version of this pper [4] ppered in the Proceedings of the Nordic Combintoril Conference in

2 G 1 G 2 Figure 1: An exmple of two grphs with the sme number of vertices, isomorphic utomorphism groups nd identicl distinguishing numbers but different number of inequivlent k-lbelings: D(G 1,k) = k 4 (k 4 1)/2 but D(G 2,k) = k 7 (k 1)/2. Let (G, φ) denote the lbeled version of G under the lbeling φ. Given two distinguishing k-lbelings φ nd φ of G, we sy tht φ nd φ re equivlent if there is some utomorphism of G tht mps (G, φ) to (G, φ ). We re interested in computing D(G, k) the number of inequivlent k-distinguishing lbelings of G which ws first considered by Arvind nd Devnur [4] nd Cheng [9] to determine the distinguishing numbers of trees. Our motivtion for studying this prmeter re s follows. First, D(G) = min{k : D(G, k) > 0} so if we cn compute D(G, k) efficiently then we cn lso determine D(G) efficiently. The usul wy of proving tht D(G) = k is to present distinguishing k -lbeling of G nd then rgue tht G hs no distinguishing lbelings tht uses k < k lbels. Counting the number of inequivlent distinguishing k-lbelings of G provides us with n ltogether different method for solving D(G). Second, when G is connected, solving for D(G, k) llows us to determine the distinguishing number of H where H = αg (i.e., H consists of α copies of G). To distinguish H, ech copy of G must be ssigned distinguishing lbeling. Additionlly, no two copies of G cn be ssigned equivlent distinguishing lbelings. Hence, D(H) = min{k : D(G, k) α}. Finlly, reserchers hve noted tht two lbels re sufficient for distinguishing mny non-rigid grphs (e.g., [2, 12, 16]). The number of inequivlent distinguishing k-lbelings of grphs provides one more level of grnulrity tht enbles us to differentite between grphs with the sme distinguishing numbers. For exmple, consider the two grphs shown in Figure 1. They hve the sme number of vertices, their utomorphism groups re isomorphic, nd they cn be distinguished with two lbels. Yet, D(G 1, k) = k 4 (k 4 1)/2 but D(G 2, k) = k 7 (k 1)/2 so with two lbels t most 120 copies of G 1 cn be distinguished compred to 64 copies for G 2. In this sense, G 1 is less symmetric thn G 2 becuse k lbels cn distinguish more copies of G 1 thn G 2 for ny k 2. To solve for D(G, k), we pply two of the most common techniques for counting the principle of inclusion-exclusion (PIE) nd recursion. We show tht when G s utomorphisms re known nd the size of its utomorphism group, Aut(G), is O(log n) where n is the number of vertices in G then strightforwrd ppliction of PIE cn determine D(G, k) efficiently. We then modify the technique so tht when Aut(G) is isomorphic to Z t (the cyclic group of order t), D t (the dihedrl group of order 2t), Z t Z 2, or D t Z 2, the vlue of D(G, k) cn be computed in time polynomil in n, t nd log k. Consequently, we re ble to prove tht if G is triconnected plnr grph then D(G, k) nd D(G) cn be determined efficiently. Next, by viewing G vi tree decomposition T G tht is mde up of G s cut vertices, seprting pirs, nd triconnected components, we show tht D(G, k) cn be determined recursively. To implement this technique efficiently for fmily of grphs, severl ingredients re necessry including efficient lgorithms for testing grph isomorphism nd finding the utomorphisms of grph s triconnected components. Since these lgorithms exist for plnr grphs, we rrive t the min result of the pper tht when G is plnr grph then D(G, k) nd D(G) cn be computed efficiently. In their introductory pper, Alberston nd Collins [3] rised the issue of determining the computtionl complexity of DIST = {(G, k) G hs distinguishing k-lbeling}. Currently, the best known result bout DIST, which is due to Russell nd Sundrm [19], is tht it belongs to AM, the set of lnguges for which there re Arthur nd Merlin gmes. This result essentilly follows from the fct tht testing grph rigidity is in AM. When G is restricted to certin grph fmilies, however, DIST cn belong to P. For exmple, distinguishing numbers of cycles, hypercubes [5, 8], nd cyclic grphs [4, 9] cn be computed efficiently. Our min result extends this further DIST belongs to P when G is 2

3 plnr grph. Our work complements tht of Fukud, et l [12] on triconnected plnr grphs where they show tht, except for seven grphs, ll grphs in this fmily hve distinguishing number t most 2. In the next section of the pper we give bsic results tht will be used throughout the pper. In Section 3, we show how the principle of inclusion/exclusion cn be used to determine D(G, k). In Section 4, we develop recursive formuls for tree decomposition of G tht eventully led to the computtion of D(G, k). We conclude in Section 5. We note tht our lgorithms for computing D(G, k) hve G nd k s input; hence, when we sy tht they re efficient, we men tht they run in time polynomil in the size of G nd log k. Additionlly, these lgorithms involve ddition nd multipliction. In cses where the numbers used re functions of k, their vlues never exceed k n, where n is the number of nodes in grph G; i.e., the numbers hve t most n log k bits. Thus, in our nlysis, we ssume ech ddition tkes O(n log k) time nd ech multipliction tkes O(n 2 log 2 k) time in the worst cse. 2 Bsic notions Suppose φ nd φ re two distinguishing lbelings of G. Since (lbeled) grph isomorphism is n equivlence reltion, we shll sy tht φ nd φ re equivlent if (G, φ) = (G, φ ); tht is, there is n utomorphism of G tht mps (G, φ) to (G, φ ). Let L(G, k) denote the set of ll distinguishing k- lbelings of G, L(G, k) the size of L(G, k), nd D(G, k) the number of equivlence clsses of L(G, k). Below, we estblish the reltionships between D(G), D(G, k) nd L(G, k). Lemm 2.1. Let G be grph nd Aut(G) its utomorphism group. (i) D(G) = min{k : L(G, k) > 0} = min{k : D(G, k) > 0}. (ii) D(G, k) = L(G, k)/ Aut(G). Proof: If there is distinguishing k-lbeling of G then the set L(G, k) must t lest hve one lbeling nd one equivlence clss. It follows tht the smllest k for which this is true must be the distinguishing number of G, proving the first prt of the lemm. To prove the second prt, note tht Aut(G) is group tht cts on L(G, k). By definition, ech φ L(G, k) is preserved by only one utomorphism in Aut(G) the identity utomorphism. Hence, ccording to the orbit-stbilizer lemm, the size of the equivlence clss of L(G, k) tht contins φ (i.e., the orbit of φ) is Aut(G). Consequently, the number of equivlence clsses of L(G, k) is L(G, k)/ Aut(G). Throughout this pper, we shll mke use of Lemm 2.1 by viewing the problem of finding grph s distinguishing number s counting problem. While it my seem tht computing D(G, k) to find D(G) requires more work thn needed, the lemm below (first proved in [9]) shows tht it does not if we need to distinguish multiple copies of G. Lemm 2.2. Let G be grph whose g connected components re G 1, G 2,..., G g. Let φ be lbeling of G. Then φ is distinguishing if nd only if the following two conditions hold: i. φ when restricted to G i is distinguishing for i = 1,...,g. ii. If G i = Gj, i j, then (G i, φ Gi ) = (Gj, φ Gj ) for every pir of i, j {1,..., g}. The following is immedite. Lemm 2.3. Let G be connected grph. If H consists of α copies of G (i.e., H = αg), then D(H) = min{k : D(G, k) α}. 2.1 Blocks, cut vertices, seprting pirs, triconnected components Let G = (V, E) be connected grph. Recll tht G is r-connected if V > r nd, for ny X V such tht X < r, removing the vertices in X from G does not disconnect G; i.e., G X remins connected. Suppose we re interested in determining if G hs some property (e.g., if it is plnr). A 3

4 common technique is to first decompose G into its blocks which re either edges or 2-connected (or biconnected) subgrphs of G nd then decompose the blocks into its triconnected components [14] which re either prllel edges (or bonds), cycles, or 3-connected grphs. 1 It is then the triconnected components which re initilly studied; the results re then ssembled to infer the properties of the blocks, which in turn infer the property of G. We shll pply this technique in Section 4 to determine D(G, k). In prticulr, we shll mke use of tree, T G, tht cptures the reltionships between the cut vertices, seprting pirs nd triconnected components of G to ssemble the informtion for computing D(G, k). A block of G is mximlly-connected subgrph of G tht does not contin cut vertex. Thus, block of G is either n edge or mximl biconnected subgrph of G. Furthermore, ny two blocks of G hve t most one vertex in common nd this vertex is cut vertex of G. The block-cut vertex grph of G is biprtite grph where one prtite set consists of b-vertices which correspond to the blocks of G, nd the other prtite set consists of c-vertices which correspond to the cut vertices of G. A b-vertex is djcent to c-vertex if nd only if the block ssocited with the b-vertex contins the cut vertex ssocited with the c-vertex. It is well known tht the the block-cut vertex grph of G is tree whose leves re b-vertices, nd so it hs unique center. Moreover, it cn be constructed in time liner in the size of G [1]. Every block of G tht is biconnected cn similrly be represented by tree vi its triconnected components nd seprting pirs. To do so, the definition of 3-connectedness nd seprting pirs hve to be extended to multigrphs. Our discussion closely follows the pper of Hopcroft nd Trjn [14]. Let B be biconnected multigrph, nd {x, y} be pir of vertices in B. The set {x, y} prtitions the edge set of B in the following wy: two edges belong to the sme clss if nd only if they lie in pth tht contins neither x nor y except possibly s endpoints. The clsses re clled the seprtion clsses of B with respect to {x, y}. If there re t lest two seprtion clsses then the pir {x, y} is seprting pir of B except when (i) there re exctly two seprtion clsses nd one clss consists of single edge, or (ii) there re exctly three clsses, ech consisting of single edge. If B is biconnected multigrph nd hs no seprting pirs then B is sid to be triconnected. Let {x, y} be seprting pir of B, nd let the seprtion clsses of B with respect to {x, y} be E 1,..., E m. An immedite consequence of the definition of seprting pirs is tht the clsses cn be divided into two groups E = k i=1 E i nd E = m i=k+1 E i so tht both E nd E hve t lest two edges. Let B = (V (E ), E {(x, y)}) nd B = (V (E ), E {(x, y)}). The grphs B nd B re clled split grphs of B with respect to (x, y) nd the edges (x, y) dded to both grphs re clled virtul edges. To split B is to replce B by two of its split grphs. Hopcroft nd Trjn suggest denoting the ith splitting opertion vi the pir {x, y} by s(x, y, i) nd lbeling the (x, y) edges dded to B nd B by i to differentite this split from other splits. Suppose B is split, its split grphs re split nd so on until there re no more splits possible. The remining grphs re clled the split components of B. Clerly, they ll must be triconnected; they cn be grouped together s follows: the triple bonds B b3, the (simple) tringles B t, nd the rest of the triconnected (simple) grphs B tg. Since there re mny wys of splitting B, the split components of B re not necessrily unique (e.g., consider four-cycle). Nonetheless, this lck of uniqueness cn be fixed by n opertion clled merge which is the reverse of split. Let B 1 = (V 1, E 1 ) nd B 2 = (V 2, E 2 ) be two split components of B tht contin virtul edge e = (x, y) lbeled i. The grph (V 1 V 2, E 1 {e} E 2 {e}) is clled the merge grph of B 1 nd B 2. To merge B 1 nd B 2 is to crete their merge grph. As before, the opertion is denoted by m(x, y, i) to differentite it from other merge opertions. So suppose the split components of B re contined in B b3 B t B tg. Merge the triple bonds in B b3 s much s possible to obtin set of bonds B b. Merge the tringles in B t s much s possible to obtin set of cycles B p. The set of grphs in B b B p B tg re clled the triconnected components of B. For exmple, cycle hs only one triconnected component itself becuse the tringles obtined by splitting the cycle cn be merged. The following hs been proven in [14]: 1 Unlike blocks, however, the triconnected components of grph need not be one of its subgrphs. 4

5 f c b d e f g h i j k n l m c e d e g h i j k j l e j b n m Figure 2: A grph nd its block-cut vertex grph. Lemm 2.4. Let B be biconnected multigrph with m B 3 edges. The totl number of edges in the split components of B is t most 3m B 6. Additionlly, the triconnected components of B re unique nd cn be found in time liner in the size of B. Lemm 2.4 implies tht the order in which the split nd merge opertions re pplied to decompose B to its triconnected components is not importnt the sme components re obtined. The biconnected multigrph B cn now be represented by its triconnected component-seprting pir grph which is biprtite grph where one prtite set consists of t-vertices tht correspond to the triconnected components of B, nd the other prtite set consists of s-vertices tht correspond to B s seprting pirs which exist s virtul edges in B s triconnected components. A t-vertex is djcent to n s-vertex if nd only if the triconnected component ssocited with the t-vertex contins the seprting pir ssocited with the s-vertex. It is esy to verify tht this biprtite grph must gin be tree, ll its leves re t-vertices nd consequently hs unique center. Moreover, becuse the triconnected components of B cn be found in liner time, the tree cn lso be constructed in liner time. Building tree-decomposition of G. Let G be (simple) connected grph. Let us now build tree decomposition of G, T G, tht incorportes the triconnected component-seprting pir grph of ech block of G into the block-cut vertex grph of G. Initilly set T G to be the block-cut vertex grph of G. Then, for ech b-vertex z whose ssocited block is B, replce z with B s triconnected componentseprting pir grph T B. Attch T B to ech neighbor y of z in the following mnner. Let be the cut vertex ssocited with y. Node is prt of one or more triconnected components nd seprting pirs of B. It is strightforwrd to check tht the vertices in T B ssocited with these components nd pirs form subtree which hs unique center becuse ll the leves of the subtree re t-vertices. Connect the center of this subtree in T B to y. For exmple, in Figure 2, the center of the block-cut vertex grph is b-vertex who hs two c-vertices s neighbors: one is ssocited with e, the other with j. When this b-vertex ws replced with the ssocited triconnected component-seprting pir grph, s shown in Figure 3, nodes e nd j re prt of ll the seprted pirs nd triconnected components. Thus, both c-vertices ssocited with e nd j re connected to the center of the triconnected component-seprting pir grph. Next, let us ssign root, r(t G ), to T G s follows. If the center of the block-cut vertex grph of G is c-vertex, this c-vertex is prt of T G. Set r(t G ) to be this c-vertex. Otherwise, the center of the block-cut vertex is b-vertex ssocited with some block B. Set r(t G ) to be the center of T B. The tree decomposition of the grph in Figure 2 is shown in Figure 3. 5

6 e j e j v v v v r e b j l b e d b d e b b d e b j k l j l j l n l n m n l l n b c d f g j e e j h i Figure 3: The tree decomposition T G of the grph in Figure 2 where r = r(t G ). The virtul edges of the triconnected components re drwn with dshed lines. 6

7 Clim 2.5. Every utomorphism of G mps the structure ssocited with r(t G ) which my be cut vertex, seprting pir, or triconnected component of G to itself. Proof: Let BC G denote the block-cut vertex grph of G. Recll tht BC G hs unique center; denote it s z. Every utomorphism of G induces n utomorphism on BC G. 2 But every utomorphism on BC G fixes z ; hence, every utomorphism of G fixes the structure ssocited with z. If z is c-vertex, r(t G ) = z nd so the clim follows. Otherwise, z is b-vertex tht is ssocited with some block B. This mens tht the ction of every utomorphism of G on B corresponds to n utomorphism of B. Now, every utomorphism of B induces n utomorphism on T B. Applying the sme rgument bove to T B, we hve tht every utomorphism of B fixes the structure ssocited with center of T B. Since r(t G ) is the center of T B, the clim follows. From here onwrds, we shll tret T G s rooted tree. For ech node v in T G, let T v denote the subtree of T G rooted t v. We define G(T v ) recursively s follows: when v is lef node of T G nd, hence, t-vertex, let G(T v ) be the triconnected component of G ssocited with v. Otherwise, v is c-, s-, or t-vertex nd hs t lest one child. When v is c-vertex, construct G(T v ) by tking the union of the grphs in {G(T w ) : w child of v}. When v is n s-vertex whose ssocited seprting pir is {x, y}, let w x nd w y denote the children of v whose ssocited cut vertices re x nd y respectively. (Note tht they my not exist.) Construct G(T v ) by first merging the grphs in {G(T w ) : w is child of v nd is t-vertex} vi {x, y} nd then ppending G(T wx ) nd G(T wy ) (if they exist) to x nd y respectively. Finlly, when v is t-vertex whose ssocited triconnected component is H, construct G(T v ) by first merging the grphs in {G(T w ) : w is child of v nd is n s-vertex} to H nd then tking the union of the resulting grph with the grphs in {G(T w ) : w is child of v nd is c-vertex}. We mke few observtions bout G(T v ). First, if w 1 nd w 2 re children of v in T G, the vertices nd edges tht G(T w1 ) nd G(T w2 ) hve in common belong to G(T v ). Thus, G(T v ) consists of the structure ssocited with v together with connected grphs hnging from it; these connected grphs re exctly from the set {G(T w ) : w is child of v}. Second, when v = r(t G ) then G(T v ) = G. Finlly, some of the G(T v ) s my not be subgrphs of G since some of them my contin one or more copies of the edge (x, y) where {x, y} is seprting pir of G but (x, y) is not n edge of G. 3 In our lter discussions, we will mostly be interested in the utomorphisms of G(T v ) tht fix cut vertex, seprting pir, or triconnected component, nd so we use Aut(G(T v ); ) to denote the set of utomorphisms of G(T v ) tht fix the structures in. For exmple, let {x, y} be seprting pir in G(T v ). The utomorphisms of G(T v ) in Aut(G(T v ); x, y) fix the vertices x nd y while those in Aut(G(T v ); xy) fix the edge (x, y). When H is triconnected component in G(T v ), the utomorphisms of G(T v ) in Aut(G(T v ); H) mp H to itself (i.e., the set V (H) to itself) nd the utomorphisms in Aut(G(T v ); H, x, y) mp H to itself nd, dditionlly, vertices x nd y to themselves. From Clim 2.5, we hve the next lemm. Lemm 2.6. Let G be connected grph nd T G its tree decomposition. Then Aut(G) = Aut(G; A) where A is the structure ssocited with r(t G ). From the construction of T G, we lso hve the next two lemms. Lemm 2.7. Let v be c-vertex in T G nd be its ssocited cut vertex. Let w be child of v in T G. The following re true: (i) if w is n s-vertex, then it is ssocited with some seprting pir {, b} nd Aut(G(T w ); ) = Aut(G(T w );, b), (ii) if w is t-vertex nd its ssocited triconnected component is H, then H contins nd Aut(G(T w ); ) = Aut(G(T w ); H, ). 2 Tht is, if π Aut(G), define f π on the set of vertices of BC G so tht f π mimics the ctions of π on G. Thus, for ech vertex z in BC G whose ssocited structure is A, let f π(z) be the vertex in BC G ssocited with the structure π(a). It is esy to verify tht f π is n utomorphism of BC G. 3 We note though tht in computing for D(G, k), we cn ignore the other copies of (x, y) s their multiplicity does not ffect ny of our computtions. 7

8 Proof: Since v nd w re djcent in T G nd v is c-vertex while w is n s- or t-vertex, there is block B tht contins cut vertex nd the structure ssocited with w. As we noted in the construction of T B, must be prt of one or more seprting pirs nd triconnected components in B, nd the vertices ssocited with these pirs nd components form subtree in T B. Let us cll this subtree T B,. Since w ws chosen so tht it is the center of T B,, the structure ssocited with w contins. By the wy the block-cut vertex grph of G is constructed, B must be the only block in G(T w ) tht contins. Hence, every utomorphism of G(T w ) tht fixes must mp the seprting pirs nd triconnected components of B tht contin to similr seprting pirs nd triconnected components. Tht is, the ctions of every utomorphism in Aut(G(T w ); ) induces n utomorphism on T B,. But T B, hs unique center w which mens tht every utomorphism in Aut(G(T w ); ) must mp the structure ssocited with w to itself. The lemm follows. Lemm 2.8. Let v be n s-vertex in T G nd {x, y} be its ssocited seprting pir. Let w be child of v in T G. If w is t-vertex whose ssocited triconnected component is H, then Aut(G(T w ); x, y) = Aut(G(T w ); H, x, y) nd Aut(G(T w ); xy) = Aut(G(T w ); H, xy). Proof: Since v nd w re djcent in T G nd v is n s-vertex while w is t-vertex, there is gin block B tht contins the structures ssocited with both vertices. By the wy the triconnected component-seprting pir grph of B is constructed, it must be the cse tht H is the only triconnected component in G(T w ) tht contins {x, y}. Hence, every utomorphism in Aut(G(T w ); x, y) must mp H to itself nd so Aut(G(T w ); x, y) = Aut(G(T w ); H, x, y). By the sme resoning, Aut(G(T w ); xy) = Aut(G(T w ); H, xy). The following lemms will lso be useful lter. Lemm 2.9. The tree T G cn be constructed in O(n 2 +nm) time where n is the number of vertices nd m the number of edges in G. Proof: Constructing G s block-cut vertex grph nd rooting it t its center tkes O(n+m) time. Creting the seprting pirs-triconnected components grph T B of block B tkes O(n B + m B ) time where n B nd m B re the number of nodes nd edges in block B. Connecting T B to T G tkes O(c B (n B + m B )) where c B is the number of cut vertices in block B. Thus, doing this for ll blocks B tkes O(n 2 + nm) time since B n B n + m nd B m B = m. Lemm Let B be block of G with n B vertices nd m B 3 edges. Let H be the set tht contins ll the triconnected components of B. For ech H H, let S H denote the set contining the seprting pirs of G in H used in the construction of T G. Then, H H S H = O(m B ) nd H H V (H) = O(m B). Proof: Suppose the split opertion ws pplied g times to B until no more splits re possible. Let H contin the resulting split components. For ech H H, define S H s in the lemm. We note tht when B is split into two components, the seprting pir used to crete the split becomes prt of both components. Tht is, ech split opertion contributes vlue of 2 to H H S H. Hence, H H S H = 2g. Now, ccording to Lemm 2.4, the totl number of edges in the split components in H is t most 3m B 6. Since split component in H hs t lest three edges, g m B 2 nd so H H S H = O(m B ). Next, notice tht V (H ) E(H ) for ech H H so H H V (H ) 3m B 6. Finlly, becuse H H S H H H S H nd H H V (H) H H V (H ), the lemm follows. Finlly, we note tht we cll T G tree decomposition of G becuse it relly is tree decomposition s defined by Robertson nd Seymour (see Chpter 12 in [11] for n introduction). Tht is, if v is node of T G nd V v contins the vertices of the structure in G ssocited with v, it should be cler from our construction tht the following re true: (i) V v V (G) for ech v, (ii) v V v = V (G), (iii) every edge of G hs two of its endpoints in some V v, nd (iv) whenever y nd z re neighbors of v then V y V z V v. In our discussion, however, it is importnt tht we keep trck of the ctul structure ssocited with v nd not just the vertices in V v. 8

9 3 Counting the distinguishing k-lbelings of grphs vi PIE Given grph G nd its utomorphisms, we begin by pplying the principle of inclusion-exclusion (or PIE) to count its distinguishing k-lbelings. Unfortuntely, the technique requires the computtion of Ω(2 Aut(G) ) terms nd so becomes imprcticl when G hs mny utomorphisms. We show how the method cn be modified when Aut(G) is isomorphic to certin groups. In prticulr, we prove tht when G is triconnected plnr grph, L(G, k), D(G, k), nd D(G) cn be computed in time polynomil in log k nd the size of G. Suppose Aut(G) = {π 0, π 1,...,π g 1 } where π 0 is the identity utomorphism. Let φ be some k-lbeling of G. We sy tht n utomorphism π i of G preserves φ if φ(v) = φ(π i (v)) for ech v of G. Clerly, π 0 preserves φ, nd if no other utomorphism of G preserves φ then φ is distinguishing k-lbeling of G. Let P Aut(G) nd N (P) denote the number of k-lbelings of G tht re preserved by ll the utomorphisms in P. Let N = (P) equl the number of k-lbelings of G tht re preserved by ll the utomorphisms in P but no others. Thus, L(G, k) = N = ({π 0 }). According to the PIE, N = ({π 0 }) = {π 0} P Aut(G) ( 1) P 1 N (P). (1) Next, we describe method for computing N (P), for ech P Aut(G). Suppose π i P. A k-lbeling φ is preserved by π i if nd only if φ ssigns the sme lbel to v nd to π i (v) for ech vertex v in G. In fct, if there is sequence of vertices v 1, v 2,..., v r such tht v j = π i (v j 1 ) for j = 2,...,r then φ must ssign ll of these r vertices the sme lbel. By extending this ide further, we rrive t the following lemm. Lemm 3.1. Let π i Aut(G) nd φ be k-lbeling of G. Let G πi be the grph whose node set is V (G) nd whose edge set consists of the pirs (v, π i (v)), v V (G). The utomorphism π i preserves φ if nd only if, for ech connected component in G πi, φ ssigns the sme lbel to ll the vertices in tht component. Consequently, let P Aut(G). The utomorphisms in P preserve φ if nd only if, for ech connected component in πi PG πi, φ ssigns the sme lbel to ll the vertices in tht component. An immedite impliction of the lemm is if πi PG πi hs t connected components nd there re k lbels vilble then N (P) = k t. We re now redy to prove the next result. Theorem 3.2. Let G be grph on n vertices nd k be positive integer. Suppose ll the utomorphisms of G re given. Then L(G, k) cn be computed in O(n 3 log 2 k + 2 Aut(G) (n Aut(G) + n logk)) time. Proof: Begin by computing nd storing the vlues k, k 2, k 3,...,k n. Set L(G, k) to 0. For ech subset P such tht {π 0 } P Aut(G), (i) construct πi PG πi nd find the number of its connected components t using bredth-first-serch nd (ii) dd ( 1) P 1 k t to L(G, k). According to eqution (1), t the end of this lgorithm the vlue of L(G, k) is the number of distinguishing k-lbelings of G. Computing the powers of k cn be done in O(n 3 log 2 k) steps. Ech itertion of the for loop tkes t most O(n Aut(G) + n log k) time where the first term in the sum ccounts for the time it tkes to construct πi PG πi nd find its connected components, nd the ltter term ccounts for dding k t to L(G, k). Since there re 2 Aut(G) 1 subsets P to consider, computing L(G, k) tkes O(n 3 log 2 k + 2 Aut(G) (n Aut(G) + n log k)) time. Corollry 3.3. Let G be grph with n vertices nd k be positive integer. Suppose ll the utomorphisms of G re given. If Aut(G) = O(log n), then L(G, k) cn be computed in time polynomil in n nd log k. The reson why implementing the PIE formul for L(G, k) cn tke exponentil time is becuse there re Ω(2 Aut(G) ) N (P) terms in the formul. Below we demonstrte tht the technique cn be 9

10 modified when Aut(G) is isomorphic to certin groups. We consider the cse when Aut(G) = Γ where (i) Γ = Z t, the cyclic group of order t, (ii) Γ = D t, the dihedrl group of order 2t, nd (iii) Γ = Z t Z 2 or D t Z 2. All will be useful when we discuss triconnected plnr grphs in the next subsection. A key feture of these results is tht Aut(G) = O(t) nd yet the number of N (P) terms tht must be computed to derive L(G, k) is polynomil in t, nd not exponentil in t. Before we proceed, we first prove the following lemm. Lemm 3.4. Let P Aut(G) nd P be the subgroup generted by P. Every k-lbeling of G preserved by ll the utomorphisms in P is lso preserved by ll the utomorphisms in P. Proof: Let φ be k-lbeling of G preserved by ll the utomorphisms in P. Let π P. Since Aut(G) is finite, we cn write π s σ r σ r 1... σ 1 where r Z + nd ech σ i P. Since ech σ i preserves φ, for ech vertex u of G, φ(u) = φ(σ 1 (u)) = φ(σ 2 (σ 1 (u))) = = φ(σ r ( (σ 2 (σ 1 (u))))). Tht is, π = σ r σ r 1... σ 1 preserves φ s well. In the subsequent discussion, when Aut(G) = Γ, we shll denote the utomorphisms of G s π σ where σ Γ, nd let π σ π σ = π σ σ. When Aut(G) = Z t. Let Z t be the cyclic group of order t nd ρ be one of its genertors. Its elements re ρ 0 (the identity), ρ, ρ 2,..., ρ t 1 where ρ i ρ j = ρ i+j mod t. Theorem 3.5. Let Aut(G) = Z t, where the prime fctoriztion of t is s π ρ of Aut(G) is given. Let P = {π ρ i : i {t/p 1, t/p 2,..., t/p s }}. Then L(G, k) = ( 1) P N (P). P P i=1 pri i. Suppose genertor Proof: To prove the theorem, we will show tht k-lbeling φ of G is distinguishing if nd only if no utomorphism in P preserves φ. One direction is obvious: if φ is distinguishing, ll non-trivil utomorphisms of G do not preserve φ. Since P contins only non-trivil utomorphisms of G, the result follows. So suppose φ is not distinguishing. It must be preserved by some π ρ j, j 0. Let g = gcd(t, j) = s i=1 pti i, where 0 t i r i. We know tht ρ g ρ j. Since j < t, we lso know tht g must divide one of the numbers in {t/p 1, t/p 2,...,t/p s }, sy t/p 1 ; i.e., ρ t/p1 ρ g. By Lemm 3.4, it follows tht if π ρ j preserves φ then π ρ g lso preserves φ, which implies tht π ρ t/p 1 does so s well. Tht is, some utomorphism in P preserves φ. Applying the PIE, L(G, k) = P P ( 1) P N (P). When Aut(G) = D t. Let D t be the dihedrl group of order 2t. If we let the genertors of D t be the rottion ρ nd reflection τ, then the elements of D t re ρ 0 (the identity), ρ 1,...,ρ t 1, τρ 0, τρ 1,...,τρ t 1, where τ 2 = ρ 0, τρ i = ρ i τ nd ρ i ρ j = ρ i+j mod t. Theorem 3.6. Let Aut(G) = D t, where the prime fctoriztion of t is s i=1 pri i. Suppose genertors π ρ nd π τ of Aut(G) re given. Let P = {π ρ i : i {t/p 1, t/p 2,..., t/p s }}. Then N = ({π ρ 0, π τρ i}) = {π τρ i } P {π τρ i } P ( 1) P 1 N (P), nd L(G, k) = t 1 ( 1) P N (P) N = ({π ρ 0, π τρ i}). (2) P P i=0 10

11 Proof: We shll first prove tht k-lbeling φ of G tht is preserved by t lest two non-trivil utomorphisms of G is lso preserved by some utomorphism in the set P = {π ρ i : i {t/p 1, t/p 2,...,t/p s }}. If one of the utomorphisms tht preserves φ is preserved by π ρ j, j 0, then by the proof of Theorem 3.5 it must lso be preserved by some utomorphism in P. If the two utomorphisms tht preserve φ re π τρ i nd π τρ j, where i < j, then π τρ i π τρ j = π ρ j i lso preserves φ. Once gin, some utomorphism in P must preserve φ. To prove eqution (2), we now consider the set of ll k-lbelings of G. Let sets A, B, nd C consist of ll k-lbelings of G preserved by π ρ 0 only, by π ρ 0 nd π τρ i for some i {0, 1,..., t 1} only, nd by some utomorphism in P respectively. Any k-lbeling of G must belong to exctly one of the three sets becuse: (i) if it is distinguishing, it belongs to set A nd if not to B C; (ii) if it is preserved by exctly one non-trivil utomorphism of G, nd it is of the form π τρ i, it belongs to set B; otherwise, it belongs to set C; (iii) finlly, if it is preserved by t lest two non-trivil utomorphisms of G, then it belongs to set C. Tht is, A B C contins ll the k-lbelings of G nd no two of them hve k-lbeling of G in common. Thus, L(G, k) = A = k n B C. By the wy we defined set B, B = t 1 i=0 N =({π ρ 0, π τρ i}). Consider k-lbeling of G tht is preserved by π τρ i. From our erlier rgument, we cn ssume tht such k-lbeling is preserved by π τρ i only or by π τρ i nd some other utomorphism in P, in ddition to being preserved by π ρ 0. According to the PIE, this mens tht N = ({π ρ 0, π τρ i}) = {π τρ i } P {π τρ i } P ( 1) P 1 N (P). Finlly, C consists of ll the k-lbelings of G preserved by t lest one of the utomorphisms in P. So, ccording to the PIE, k n C = P P ( 1) P N (P). Hence, L(G, k) = P P ( 1) P N (P) t 1 i=0 N =({π ρ 0, π τρ i}), which proves eqution (2). Exmple. Consider the cycle on n vertices C n, where n is prime number. Then Aut(C n ) = D n nd P = {π ρ }. To solve for L(C n, k), we need the following vlues: N ( ), N (π ρ ), N ({π τρ i, π ρ }) nd N ({π τρ i}). Every k-lbeling of C n should be counted in N ( ) so N ( ) = k n. To solve for N (π ρ ), recll tht we considered G πρ which is grph tht hs only one component. Hence, N (π ρ ) = k. Similrly, N ({π τρ i, π ρ }) = k. Finlly, G πτρ i consists of (n + 1)/2 components since ny reflection of C n fixes one vertex v nd mps the equidistnt vertices from v to ech other. Thus, N ({π τρ i}) = k (n+1)/2. From eqution (2), n 1 ( L(C n, k) = N ( ) N ({π ρ }) N ({π τρ i}) N ({π τρ i, π ρ }) ) i=0 = k n k nk (n+1)/2 + nk = k n nk (n+1)/2 + (n 1)k = k(k (n 1)/2 1)(k (n 1)/2 (n 1)). Consequently, D(C n, k) = k(k (n 1)/2 1)(k (n 1)/2 (n 1))/2n. When n = 5, for exmple, D(C 5, 1) = D(C 5, 2) = 0 but D(C 5, 3) = 12 so D(C 5 ) = 3. When Aut(G) = Z t Z 2 or D t Z 2. We stte the following theorem without proof becuse the rguments re just extensions of those in Theorems 3.5 nd 3.6. Theorem 3.7. Suppose the prime fctoriztion of t is s i=1 pri i, the group Z t Z 2 = {(ρ i, σ j ), i {0, 1,..., t 1}, j {0, 1}} nd the group D t Z 2 = {(ρ i, σ j ), (τρ i, σ j ), i {0, 1,..., m 1}, j {0, 1}}. When t is odd, set P0 = {π (ρ 0,σ)}; otherwise, set P0 = {π (ρ 0,σ), π (ρ,σ)}. Let P = t/2 P0 {π (ρ i,σ 0 ) : i {t/p 1, t/p 2,...,t/p s }}. (i) When Aut(G) = Z t Z 2, L(G, k) = P P ( 1) P N (P). 11

12 (ii) When Aut(G) = D t Z 2, nd for b = 0 or 1, N = ({π (ρ 0,σ 0 ), π (τρi,σ b )}) = {π (τρ i,σ b ) } P {π (τρ i,σ b ) } P ( 1) P 1 N (P) nd L(G, k) = ( 1) P N (P) P P 1 t 1 N = ({π (ρ 0,σ 0 ), π (τρ i,σ b )}). b=0 i=0 Remrk: Since the number of prime fctors of t is O(log t), the number of N (P) terms in the formul for computing L(G, k) is O(t) when Aut(G) = Z t or Z t Z 2, nd O(t 2 ) when Aut(G) = D t or D t Z When G is triconnected plnr grph Wht is interesting bout the fmily of triconnected plnr grphs is tht the utomorphism groups of the grphs re only of limited kinds. Fct 3.8. [18] Let G be triconnected plnr grph. The utomorphism group of G is isomorphic to subgroup of one of the following groups: A 4, A 5, S 4, A 4 Z 2, A 5 Z 2, S 4 Z 2, Z t, D t, Z t Z 2, D t Z 2, for some integer t. Since subgroup of dihedrl group is cyclic group or dihedrl group, clerly the subgroups of D t Z 2 re cylic, dihedrl or isomorphic to Z t Z 2 or D t Z 2 where t t. In other words, the utomorphism group of triconnected plnr grph is either bounded by constnt or it is isomorphic to one of four groups only. Additionlly, becuse triconnected plnr grphs hve only unique embeddings on the plne up to equivlence 4, finding ll their utomorphisms cn lso be done efficiently. We sketch one such method next. Let G be triconnected plnr grph with n vertices nd m edges. Let e = (u, v) be n edge of G. Let us designte its direction s being from u to v nd one of the fces F tht it borders s its right fce. Crete copy of G, G e,f, which specilly mrks e nd its direction, nd fce F. For ny edge e = (u, v ) whose direction nd right fce F is fixed, crete n nlogous grph G e,f, nd using plnr grph isomorphism testing lgorithm determine if G e,f nd G e,f re isomorphic (where the mrked edge nd fce of G e,f re mpped to the mrked edge nd fce of G e,f ). If so, then there is n utomorphism of G tht mps e to e nd F to F ; moreover, by visiting the fces of G e,f nd G e,f in the sme order, the rest of π cn be determined in time liner in the size of G. Since there is liner time isomorphism testing lgorithm for plnr grphs [15], ech itertion of the for loop tkes O(n) time. And since there re O(m) itertions then in O(nm) = O(n 2 ) time ll the utomorphisms of G cn be determined. Furthermore, becuse ech edge hs two directions nd two fces bordering it, the lgorithm bove lso shows tht Aut(G) 4m = O(n) when G is triconnected plnr grph. To solve for L(G, k) for triconnected plnr grphs, we do the following: if Aut(G) 5!, use Theorem 3.2. Otherwise, determine if Aut(G) is cyclic, dihedrl, isomorphic to direct product of cyclic group nd Z 2, or to direct product of dihedrl group nd Z 2. If Aut(G) is cyclic or dihedrl, pply Theorems 3.5 or 3.6 respectively; otherwise, pply Theorem 3.7. Theorem 3.9. Let G be n n-vertex triconnected plnr grph. Computing L(G, k) nd D(G, k) cn be done in O(n 3 log 2 k +n 3 log n+n 3 log k) time. Consequently, computing D(G) tkes O(n 3 log 3 n) time. 4 A triconnected plnr grph cn hve two plnr embeddings one of which is mirror imge of the other. 12

13 Proof: As we stted erlier, if G hs t most 5! utomorphisms, we use Theorem 3.2 to solve for L(G, k) nd D(G, k). Otherwise, we need to determine which of the four groups Aut(G) is isomorphic to. In prticulr, Aut(G) flls into CASE i where i = 1 if the group is cyclic, i = 2 if the group is isomorphic to Z t Z 2 for some t, i = 3 if the group is dihedrl, nd i = 4 if the group is isomorphic to D t Z 2 for some t. We note tht there is some overlp in the four cses becuse if t is odd, Z t Z 2 = Z2t nd D 2t Z 2 = D4t. Thus, when we sy tht Aut(G) belongs to CASE 2 or 4, we shll ssume tht t is even. We describe our lgorithm T riconnectcount(g, k) in Figure 3.1. In the first prt of our lgorithm, we determine the cse which Aut(G) belongs to by considering the order of ech element in Aut(G). It is esy to verify the following fcts: (i) if Aut(G) hs n element with order Aut(G) it must be cyclic, (ii) if Aut(G) hs only three elements with order 2 (nd 3 < Aut(G) /2) then it belongs to cse 2, (iii) if Aut(G) hs between Aut(G) /2 nd Aut(G) /2+1 of its elements with order 2, it belongs to cse 3. Once the pproprite cse for Aut(G) is determined, we set the vlue of t. The second prt of the lgorithm begins by computing the prime fctors of t, finding n element π Aut(G) such tht the order of π is t, nd then computing P = {π i : i {t/p 1, t/p 2,...,t/p s }}. If Aut(G) is cyclic or dihedrl, P is indeed the one needed in Theorems 3.5 nd 3.6 respectively to compute L(G, k). In cses 2 nd 4, two more elements re missing in P. To understnd wht they re, we note tht since t is even π would be of the form π (ρ,σb ) where b = 0 or 1, nd ρ nd σ re genertors of Z t nd Z 2 respectively. If we set p 1 = 2, then (π (ρ,σ b )) t/2 = π (ρ t/2,σ 0 ) or π (ρ t/2,σ 1 ), nd (π (ρ,σb )) t/pi = π (ρ t/p i,σ 0 ) for i = 2,...,s. At this point, the two missing elements in P hve order 2; they cn be distinguished from the other elements of Aut(G) with order 2 becuse they commute with every other element of Aut(G) (i.e., they belong to the center of Aut(G)), wheres the others do not. By updting P, we now obtin the pproprite P in Theorem 3.7. Finlly, for cses 3 nd 4, we plce ll elements of Aut(G) with order 2 not in P into set T. It is esy to check tht the rest of the lgorithm computes L(G, k) correctly since they follow directly from the theorems we hve estblished. Computing nd storing the powers of k tkes O(n 3 log 2 k) time. Finding ll the utomorphisms of G tke O(n 2 ) time. It is esy to verify tht in the rest of the lgorithm, the bottleneck is in computing the vlue of L(G, k) when Aut(G) > 5!. Applying the sme nlysis we used in Theorem 3.2, nd noting tht P = O(log t) nd T = O(t), computing L(G, k) tkes O(t 2 (n log t + n log k)) time. Finlly, becuse G is triconnected grph Aut(G) = O(n) so t = O(n). Hence, the totl runtime of TriconnectCount(G, k) is O(n 3 log 2 k + n 3 log n + n 3 log k). Once we hve the vlue for L(G, k), we simply divide it by Aut(G) to determine D(G, k). To find D(G), do binry serch over the rnge [1, n] to determine the smllest k for which D(G, k) > 0 to find D(G). This dds n extr log n fctor to the runtime of TriconnectCount(G, k). 4 Computing D(G, k) vi recursion In this section, we shll generlize the recursive technique (discovered independently by Arvind nd Devnur[4] nd by Cheng[9]) tht ws used to compute the distinguishing numbers of trees. The min ide behind the technique is quite simple. Let T be tree rooted t r. Let T v denote the subtree of T rooted t vertex v. Strt by setting D(T v, k) = k for ech lef v since single node hs k distinguishing k-lbelings. Then, for i = height(t) 1 to 0, do the following: for ll nodes v t depth i, compute D(T v, k) bsed on the vlues computed for D(T w, k), w child of v in T. Thus, t the end of the lgorithm D(T r, k), which equls D(T, k), is determined. To pply the bove technique to connected grph G, we will view G s rooted tree using the tree decomposition T G described in Section 2.1. Additionlly, we will lso consider generlized version of the distinguishing k-lbelings of grph which we shll define shortly. Finlly, we will develop recursive formuls tht relte the number of (generlized) inequivlent distinguishing k-lbelings of G(T v ) with those of G(T w ), w child of v in T G. Let Γ be subgroup of Aut(G). We sy tht lbeling φ of G is Γ-distinguishing if no non-trivil 13

14 T riconnectcount(g, k) Input: A triconnected plnr grph G with n vertices, positive integer k. Output: The vlue of L(G, k). Compute nd store the vlues k,k 2,k 3,...,k n. Find ll the utomorphisms of G. If Aut(G) 5! L(G,k) {π 0 } P Aut(G) ( 1) P 1 N (P) return(l(g, k)) else compute the order of ech utomorphism π Aut(G) if there is n utomorphism whose order is Aut(G) CASE 1, t Aut(G), else if there re only 3 utomorphisms with order 2 CASE 2, t Aut(G) /2, else if there re between Aut(G) /2 nd Aut(G) /2 + 1 elements with order 2 CASE 3, t Aut(G) /2, else CASE 4, t Aut(G) /4. Compute the prime fctors of t: p 1,p 2,...,p s. Find n utomorphism π Aut(G) whose order is t. Compute P = {π i : i {t/p 1,t/p 2,...,t/p s }}. If CASE = 2 or 4 dd to P the two utomorphisms of G which belong to the center of Aut(G) not yet in P. If CASE = 3 or 4 let T consist of ll utomorphisms in Aut(G) tht is not in P whose order is 2. L(G,k) P P ( 1) P N (P). If CASE = 1 or 2 return(l(g, k)) else while T pick π T nd delete π from T L(G,k) L(G,k) {π } P {π } P ( 1) P 1 N (P) return (L(G,k)). Figure 4: The lgorithm for computing the number of distinguishing k-lbelings of triconnected plnr grph. 14

15 utomorphism in Γ preserves φ, nd tht two lbelings φ nd φ of G re equivlent with respect to Γ if some utomorphism in Γ mps (G, φ) to (G, φ ). Let L(G, k; Γ) be the set consisting of the Γ- distinguishing k-lbelings of G, L(G, k; Γ) be the size of L(G, k; Γ), nd D(G, k; Γ) be the number of equivlence clsses of L(G, k; Γ) with respect to Γ. When Γ = Aut(G; ) s defined in Section 2.1, we shll refer to L(G, k; Γ), L(G, k; Γ) nd D(G, k; Γ) s L(G, k; ), L(G, k; ), nd D(G, k; ) respectively. Finlly, when (x, y) is n edge of G, we will t times differentite between the cse when k-lbeling of G ssigns x nd y the sme or different colors. When we do so, we will plce subscript next to L, L, nd D; the subscript is 1 if x nd y re ssigned the sme color nd is 2 otherwise. Thus, L 1 (G, k; xy) consists of ll k-lbelings of G in L(G, k; xy) tht ssigned x nd y the sme color, etc. It is esy to verify tht the following version of Lemm 2.1 remins true: Lemm 4.1. Let G be grph nd Γ be subgroup of Aut(G). Then D(G, k; Γ) = L(G, k; Γ)/ Γ. Given connected grph G, we showed in Section 2.1 how to construct tree decomposition of G, T G. The construction strted with G s block-cut vertex grph. Ech b-vertex whose ssocited block is B is then replced with B s triconnected component-seprting pir grph T B nd then connected to the rest of block-cut vertex grph. Thus, T G is mde up of c-, s-, nd t-vertices which represent the cut vertices, seprting pirs nd triconnected components of G. We shll now describe recursive formuls for D(G(T v ), k; ) bsed on the type of vertex v is in T G. Theorem 4.2. Let v be c-vertex in T G nd be the cut vertex in G ssocited with v. Suppose when ll the grphs in G = {G(T w ) : w is child of v in T G } re fixed t, there re g isomorphic clsses nd the ith isomorphic clss contins m i copies of the connected grph G i ; i.e., G = m 1 G 1 m 2 G 2... m g G g. Then g ( ) D(Gi, k; )/k D(G(T v ), k; ) = k. i=1 Proof: By the wy T G ws constructed, if G = m 1 G 1 m 2 G 2... m g G g then G(T v ) is mde up of g i=1 m i connected components ll hnging from vertex. It is esy to verify tht φ is lbeling in L(G(T v ), k; ) if nd only if φ ssigns inequivlent lbelings from L(G i, k; ) to the m i copies of G i, i = 1,...,g nd the lbels ssigned to vertex by ll the lbelings re the sme. This mens tht n equivlence clss of L(G(T v ), k; ) is defined by (i) the lbel ssigned to nd (ii) the set of m i equivlence clsses from L(G i, k; ) tht contin the lbelings of the m i copies of G i, i = 1,...,g. There re k possible lbels for. Once the lbel for is chosen sy l, there re D(G i, k; )/k different equivlence clsses of L(G i, k; ) which ssign vertex the sme lbel. This is so becuse the number of equivlence clsses of L(G i, k; ) where is ssigned the lbel l must be the sme for every possible vlue of l. It follows tht there re ( D(G i,k;)/k) m i different sets of mi equivlence clsses tht cn contin the lbelings ssigned to the m i copies of G i for i = 1,...,g. By the product rule of counting, the theorem is estblished. The next two theorems del with the cse when v is n s-vertex. Theorem 4.3. Let v be n s-vertex in T G nd {x, y} be the seprting pir ssocited with v. If it exists, let w x denote the child of v tht is c-vertex ssocited with x. Similrly, if it exists, let w y denote the child of v tht is c-vertex ssocited with y. Suppose when ll the grphs in G = {G(T w ) : w is t-vertex nd child of v in T G } re fixed t x nd y, there re g isomorphic clsses nd the ith isomorphic clss hs m i copies of the connected grph G i ; i.e. G = m 1 G 1 m 2 G 2... m g G g. Then D(G(T v ), k; x, y) equls k 2 mx{d(g(t wx ), k; x)/k, 1} mx{d(g(t wy ), k; y)/k, 1} m i g ( D(Gi, k; x, y)/k 2 ). Proof: Once gin, it is esy to verify tht φ L(G(T v ), k; x, y) if nd only if φ ssigns inequivlent lbelings from L(G i, k; x, y) to the m i copies of G i, i = 1,...,g nd the lbels ssigned to x nd to i=1 m i 15

16 1 1 2 x y 1 2 Figure 5: In this grph, (x,y) is seprting pir. Notice tht the lbeling of the 5-cycle on the left does not destroy the utomorphism of the 5-cycle tht flips the grph long the edge (x,y) but the lbeling for the entire grph is distinguishing. y by ll the lbelings re the sme. Thus, n equivlence clss of L(G(T v ), k; x, y) is defined by (i) the lbels ssigned to x nd y, (ii) the equivlence clsses of L(G(T wx ), k; x) nd L(G(T wy ), k; y) tht contin the lbelings of G(T wx ) nd G(T wy ) respectively, nd (iii) the set of m i equivlence clsses of L(G i, k; x, y) tht contin the lbelings of the m i copies of G i, i = 1,...,g. There re k 2 lbels vilble for x nd y. Once the lbels re chosen sy l x nd l y, the number of equivlence clsses of L(G i, k; x, y) where x nd y re ssigned the sid lbels is D(G i, k; x, y)/k 2. This is so becuse in ny lbeling in L(G i, k; x, y), the lbels of the vertices in G i other thn x nd y re the ones tht ctully destroy the non-trivil utomorphisms of Aut(G i ; x, y). Consequently, the number of lbelings nd the number of equivlence clsses of L(G i, k; x, y) with x nd y ssigned l x nd l y must be the sme regrdless of the pir (l x, l y ). Of the D(G i, k; x, y)/k 2 equivlence clsses of L(G i, k; x, y) tht re being considered, m i must be chosen to contin the lbelings of the m i copies of G i. Similrly, the number of equivlence clsses of L(G(T wx ), k; x) where x is ssigned the lbel l x nd the number of equivlence clsses of L(G(T wy ), k; y) where y is ssigned the lbel l y re D(G(T wx ), k; x)/k nd D(G(T wy ), k; y)/k respectively. By the product rule of counting, the theorem is estblished. We will lso need to compute D(G(T v ), k; xy). Unlike our previous chrcteriztions, however, it is not necessrily the cse tht when φ L(G(T v ), k; xy) then φ ssigned inequivlent lbelings from L(G i, k; xy) to ech copy of G i. Figure 4 shows one such exception. Our pproch this time is to consider the equivlence clsses of L(G(T v ), k; x, y) nd count those tht do not belong to L(G(T v ), k; xy). Consider n rbitrry grph H with n edge (x, y) nd suppose Aut(H; xy) Aut(H; x, y). Let Aut(H; x y, y x) denote the set of utomorphisms of H tht mp x to y nd y to x. Notice tht Aut(H; xy) is the disjoint union of Aut(H; x, y) nd Aut(H; x y, y x). Moreover, Aut(H; x, y) is subgroup of Aut(H; xy) nd Aut(H; x y, y x) is coset of Aut(H; x, y). Suppose φ 1 nd φ 2 belong to the sme equivlence clss in L(H, k; x, y). If some utomorphism in Aut(H; x y, y x) preserves φ 1, then it is esy to verify tht nother utomorphism in Aut(H; x y, y x) lso preserves φ 2. Tht is, for ech equivlence clss in L(H, k; x, y), either ll the lbelings do not destroy some utomorphism of Aut(H; x y, y x) (nd so do not belong to L(H, k; xy)), or ll do (nd so belong to L(H, k; xy)). Let B(H, k; x, y) be the set tht contins ll equivlence clsses of L(H, k; x, y) whose lbelings do not belong to L(H, k; xy), nd denote its size s B(H, k; x, y). Our discussion will focus on computing B(H, k; x, y) becuse this is the number of bd equivlence clsses of L(H, k; x, y) in tht they do not crry over s equivlence clsses of L(H, k; xy). Suppose π Aut(H; x y, y x). It is esy to verify tht (i) when ll the lbelings in n equivlence clss of L(H, k; x, y) destroy ll the utomorphisms in Aut(H; xy) then π mps these lbelings to the lbelings of nother equivlence clss of L(H, k; x, y); (ii) on the other hnd, when ll the lbelings in n equivlence clss of L(H, k; x, y) do not destroy the utomorphisms in Aut(H; xy) then π mps these lbelings to themselves. In other words, under the ction of Aut(H; xy) the equivlence clsses of L(H, k; x, y) either get pired up or sty singleton. The ones tht get pired up re precisely the equivlence clsses of L(H, k; xy); i.e., ech equivlence clss of L(H, k; xy) is mde up of two 16