Hypercbe problems Lectre 5 Noember 6, 2012 Lectrer: Petr Gregor Scribe by: Kryštof Měkta Updated: Noember 22, 2012 1 Partial cbes A sbgraph H of G is isometric if d H (, ) = d G (, ) for eery, V (H); that is, H preseres all distances from G. Clearly, eery isometric sbgraph is an indced sbgraph. To see that the conerse implication is not tre, consider a path P 4 in the cycle C 5. Figre 1: A path P 4 in the cycle C 5 is indced bt not isometric sbgraph. A graph G is a partial cbe if it has an isometric embedding into Q n for some n. That is, G is isomorphic to an isometric sbgraph of Q n for some n. The smallest sch n (if it eists) is the isometric dimesion of G, denoted by dim I (G). Obseration 1 dim I (T ) = n 1 for eery tree T on n ertices. dim I (C 2n ) = n for eery n 2, with isometric embedding sch that eery direction is sed eactly twice, on antipodal edges. Eamples of partial cbes: the Desarges graph, the permtahedron, benzenoid graphs, linear etension graphs, hyperplane arrangement graphs, Fibonacci cbes, see Figre 2. 2 Djokoić-Winkler relation Let G = (V, E) be a bipartite graph. For E let W = { V ; d(, ) < d(, )} and W = { V ; d(, ) < d(, )}. Since G is bipartite, W W is a partition of V. Obseration 2 All ertices on eery shortest -path are in W (that is, I(, ) W ) if W. The Djokoić-Winkler 1 relation, denoted by Θ, is defined as follows. Definition 3 eθf for edges e = and f if f joins a erte in W with a erte in W. 5-1
111111 00011 00111 111110 011111 01001 01011 01010 01101 10011 00101 01110 10010 00110 10110 111100 110100 011010 110101 110111 100111 001111 000111 10001 10101 111000 011000 001010 100101 001011 11001 11010 01100 10100 110000 000011 001000 11000 11100 100000 (a) 000000 153624 (b) 100001 000001 153264 135624 315624 152364 153246 135264 315264 152346 125364 135246 315246 132564 312564 125346 123564 132546 312546 123546 (c) 123456 132456 312456 (d) Figre 2: Eamples of partial cbes: (a) the Desarges graph (also known as the generalized Petersen graph P (10, 3) or the middle leel graph of Q 5 ), (b) the permtahedron of order 4 (also known as the trncated octahedron), (c) the linear etension graph of a (now secret) poset, (d) a benzenoid graph. From Obseration 2 we obtain the following. Obseration 4 For eery bipartite graph G, the relation Θ is refleie and symmetric, if edges e, f both belong to a same shortest path in G, then e Θf. To see that the relation Θ does not hae to be transitie, consider K 2,3 on Figre 3(b). 1 The definition gien here is de to Djokoić [8]. Winkler [17] defined another relation which in bipartite graphs coincides with this definition. Hence it is commonly called Djokoić-Winkler relation. 5-2
a) W W b) W f 2 W f f 1 Θ e e Figre 3: (a) a partition of V to W and W, (b) K 2,3 with eθf 1, eθf 2, bt f 1 Θf 2. 3 Characterization of partial cbes A set S V is cone if I(, ) S for eery, S where I(, ) is the interal between and. That is, with eery two ertices S contains eery shortest path between them. Theorem 5 (Djokoić [8], Winkler [17]) Let G = (V, E) be a connected graph. following statements are eqialent. The 1. G is a partial cbe, 2. G is bipartite and W, W are cone for eery E, 3. G is bipartite and Θ is transitie (and conseqently, an eqialence). Proof 1) 2) Let f : V V (Q n ) be an isometric embedding. The sbgraph f(g) of Q n is clearly bipartite which implies that G is bipartite. If E then clearly f() f() = e i for some i. Assme withot loss of generality that f() i = 0. Since f(g) is an isometric sbgraph of Q n and distances in Q n are measred by the Hamming distance it follows that f() i = 0 for eery W. A shortest path between f( 1 ), f( 2 ) for 1, 2 W clearly contains at most one edge of each direction. Ths, it does not contain any edge in the direction i and all ertices on the shortest path hae the i-th coordinate eqal to 0. This means that the shortest path between f( 1 ) and f( 2 ) stays in f(w ) so the same is tre for its preimage - the shortest path between 1, 2 W in G. 2) 3) Let ()Θ(y), say W and y W. Symmetry of Θ implies that W y and W y. We claim that W = W y. Sppose w W \ W y. Then is on a shortest w-path and w W y. The coneity of W y and the fact that w, W y imply that W y, a contradiction. The claim implies that Θ is transitie. 5-3
3) 1) We proe this implication by constrcting an isometric embedding f of G to Q n where n is the inde of Θ, i.e. n = E / Θ. Let e i = i i be a representatie of the i-th class of Θ. The embedding f : V (G) Z n 2 is defined as follows. f() i = { 0 if Wi i 1 if W i i Note that eθe if and only if f(e) and f(e ) are of the same direction in Q n. For distinct, y V a shortest y-path has edges from distinct Θ-classes which means that f() f(y), so f is injectie. By the same reason, f maps shortest paths from G to shortest paths in Q n. Ths, f is an isometric embedding. Note that from the proof it follows that dim I (G) is the inde of Θ. Since the relation Θ can be fond and tested for transitiity in a polynomial time, we obtain the following. Corollary 6 Partial cbes can be recognized in polynomial time. In fact, they can be recognized in O(mn) time and O(n 2 ) space where n = V and m = E [1]. 4 Median graphs A set of medians for a triple of ertices, y, z is defined by Med(, y, z) = I(, y) I(y, z) I(z, ). A graph G is a median graph if Med(, y, z) = 1 for eery, y, z V (G). That is, eery triple has a niqe median, which is then denoted by med(, y, z). The class of median graphs incldes all trees, grids, hypercbes (by the following proposition). We will see that median graphs are in some sense graphs between trees and hypercbes. They occr in many at first seemingly nrelated areas sch as the stdy of soltions of 2-SAT [10], stable configrations of nonepansie networks [10], stable matchings in the well-known stable roommate problem [7], median semilattices [4]. Proposition 7 Q n is a median graph for eery n 1 with med(, y, z) = maj(, y, z) where maj is the coordinate-wise majority fnction. Proof If two ertices agree in a coordinate, say i = y i, then eery m Med(, y, z) has m i = i = y i since m belongs to a shortest y-path, so m = maj(, y, z). On the other hand, if m = maj(, y, z) then i = y i implies m i = i = y i, so m lies on a shortest path between eery pair of ertices. Eery median graph is bipartite. Indeed, for a contradiction consider a shortest odd cycle with a erte antipodal to an edge yz. Then Med(, y, z) =. Theorem 8 Eery median graph G = (V, E) is a partial cbe. 5-4
W y W z P y m P 1 m P 2 z P 3 U U Figre 4: (a) The sets U and U, (b) a shortest -path P in U and adjacent ertices from U, (c) the shortest paths between and y, and, y and. Proof For E let U = { W has a neighbor in W } and similarly U. Claim 9 If U then eery shortest -path belongs to U (that is, I(, ) U ). Let z be a neighbor of on a shortest -path P, see Figre 4(b). Clearly, d(z, y) = 2 and d(z, ) = d(, ) = d(y, ). Then m = med(z, y, ) mst be a neighbor of z lying on a shortest y-path. Therefore z U. Then we contine this argment for the net erte z on P ntil we reach. Claim 10 U indces an isometric sbgraph. Let, y U and m = med(, y, ). Let P 1 be a shortest m-path, P 2 a shortest ym-path and P 3 a shortest m-path, see Figre 4(c). By Claim 4, P 1 P 3 and P 2 P 3 are in U which means that P 1 P 2 also lies in U (i.e. d U (, y) = d G (, y)). Therefore, W and similarly W are cone and we may apply Theorem 5. Not eery partial cbe is a median graph, see Figre 5. y y z z Figre 5: Two eamples of partial cbes that are not median graphs 5 Mlder s cone epansion Let G = (V, E) be a connected graph and S V be a cone set. The epanded graph G = ep(g, S) is obtained as follows: for eery erte in S add a copy and join and by an edge, 5-5
insert an edge between new ertices wheneer is an edge of S. The set of new ertices in denoted by S (a copy of S), see Figre 6. G S S Figre 6: The epansion procedre. Since a sbgraph of a median graph indced by a cone set is clearly median, by considering all cases for triples in G it can be shown that if G is a median graph, then G is also a median graph. Interestingly, eery median graph can be obtained by this procedre 2 starting from a single erte. This shows s a tree-like strctre of median graphs. Theorem 11 (Mlder [14]) A graph G is a median graph if and only if it can be obtained from a single erte by a seqence of cone epansions. For a proof we refer to [14]. Figre 7: Eamples of cone epansions. 6 Eler-type formla Recall that ertices and edges can be seen as 0-faces and 1-faces, respectiely. Ths the following relation can been regarded as an Eler-type formla. Theorem 12 (Škrekoski [16]) Let G be a median graph, k = dim I (G), and let q i be the nmber of sbcbes of dimension i in G. Then, ( 1) i q i = 1 and k = ( 1) i iq i. 2 Originally, the cone epansion is described in a more general way. Howeer, it can be shown that this simplified ersion sffices. 5-6
Proof Let G = ep(g, S), q i be the nmber of copies of Q i in G, qi S be the nmber of copies of Q i in S. From the epansion procedre, obsere that q i = q i + qs i + qi 1 S for eery i 0 (where we define q 1 S = 0). Then, ( 1) i q i = ( 1) i q i + ( 1) i qi S + ( 1) i+1 qi S = 1 (1) i 1 }{{} = 0 since q 1 S = 0 by indction for G. Frthermore, applying indction for G and (1) for S gies s k = k + 1 = ( 1) i iq i + ( 1) i q S i = = ( 1) i iq i ( 1) i (iqi S (i + 1)qi S ) = = ( 1) i iq i ( 1) i iq S i ( 1) i iqi 1 S = i 1 ( 1) i q i. 7 Open problems Problem 1 (Eppstein [9]) Find a nonplanar cbic partial cbe other than the Desarges graph. (Here, cbic means 3-reglar.) Conjectre 13 (Brešar et al. [5]) Eery cbic partial cbe is Hamiltonian. Problem 2 Characterize Hamiltonian partial cbes (median graphs). A partial cbe G = (V, E) is Θ-gracefl if there is a bijection f : V {0,..., V 1} sch that for eery e 1, e 2 E : where f () = f() f(). e 1 Θe 2 if and only if f (e 1 ) = f (e 2 ) Conjectre 14 (Brešar, Klažar [6]) Eery partial cbe is Θ-gracefl. Remark This conjectre implies the gracefl tree conjectre. 8 Notes Median graphs were first stdied by Aann [2], Nebeský [15], and independently by Mlder [14]. There is an etensie literatre on median graphs, see [3, 13] for sreys. A good starting reference is also the book 3 of Imrich and Klažar [12]. 3 The second edition [11] is already aailable. 5-7
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