Characterization of reducible hexagons and fast decomposition of elementary benzenoid graphs

Transcription

1 Discrete Applied Matematics 156 (2008) Caracterization of reducible exagons and fast decomposition of elementary benzenoid graps Andrej Taranenko, Aleksander Vesel 1 Department of Matematics and Computer Science, FNM, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia Received 7 May 2007; received in revised form 1 August 2007; accepted 12 August 2007 Available online 20 September 2007 Abstract A benzenoid grap is a finite connected plane grap wit no cut vertices in wic every interior region is bounded by a regular exagon of a side lengt one. A benzenoid grap G is elementary if every edge belongs to a 1-factor of G. A exagon of an elementary benzenoid grap is reducible, if te removal of boundary edges and vertices of results in an elementary benzenoid grap. We caracterize te reducible exagons of an elementary benzenoid grap. Te caracterization is te basis for an algoritm wic finds te sequence of reducible exagons tat decompose a grap of tis class in O(n 2 ) time. Moreover, we present an algoritm wic decomposes an elementary benzenoid grap wit at most one pericondensed component in linear time Elsevier B.V. All rigts reserved. Keywords: Benzenoid grap; 1-factor; Reducible exagon; Reducible face decomposition 1. Introduction A benzenoid grap is a finite connected plane grap wit no cut vertices in wic every interior region is bounded by a regular exagon of a side lengt one. A coronoid is a connected subgrap of a benzenoid grap suc tat every edge belongs to at least one exagon and it contains at least one non-exagonal interior face. A benzenoid grap G is catacondensed if any triple of exagons of G as empty intersection, oterwise it is pericondensed, cf. Fig. 1. A grap G is called bipartite if it is connected and its vertex set can be divided in two disjoint sets V 1 and V 2 suc tat V 1 V 2 = V (G) and no two vertices from te same set are joined by an edge. Every benzenoid grap is clearly bipartite. A peak (valley) of a benzenoid grap is a vertex tat is above (below) all its first neigbors. Trougout tis paper all benzenoid graps considered are drawn so tat an edge-direction is vertical and te peaks are colored black (see Fig. 1). A matcing of a grap G is a set of pairwise independent edges. A matcing is a 1-factor if it covers all te vertices of G. Te fact tat a Kekulé structure of a conjugated molecule is in a one-to-one correspondence wit a 1-factor of te underlying molecular grap is well and long known. In particular, te skeleton of carbon atoms in a benzenoid ydrocarbon is a benzenoid grap. Te number of 1-factors/Kekulé structures of a benzenoid grap can be used to forecast some pysico-cemical properties of te underlying compound, terefore 1-factors of benzenoid graps ave addresses: andrej.taranenko@uni-mb.si (A. Taranenko), vesel@uni-mb.si (A. Vesel). 1 Supported by te Ministry of Science of Slovenia under te Grant 0101-P X/$ - see front matter 2007 Elsevier B.V. All rigts reserved. doi: /j.dam

2 1712 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) Fig. 1. A pericondensed, coronoid and catacondensed grap, from left to rigt, respectively. been studied extensively [3]. On te oter and, some problems involving 1-factors are extended to some oter, for te most part more general classes of graps, suc as exagonal, bipartite, and plane bipartite graps. Among many different topics studied on tis classes of graps we briefly mention counting te number of 1-factors [2,18,20], finding te 1- factors [5], te binary coding of 1-factors [9] and te concept of te resonance graps (also called te Z-transformation graps) [1,8,10,16,19]. A bipartite grap G is called elementary if G is connected and every edge belongs to a 1-factor of G. Elementary components of G are te components of te grap obtained from G by removing tose edges of G tat are not contained in any 1-factor. It is well known tat catacondensed benzenoid graps are elementary. An important property of elementary bipartite graps is te bipartite ear decomposition [14]. In[22], Zang and Zang evolved tis concept and presented te so-called reducible face decomposition (RFD). Tis decomposition can serve as a construction metod for elementary bipartite graps. Let x be an edge. Join its end vertices by a pat P 1 of odd lengt (first ear). Ten proceed inductively to build a sequence of bipartite graps as follows: if G r 1 = x + P 1 + P 2 + +P r 1 as already been constructed, add te rt ear P r (of odd lengt) by joining any two vertices of different colors in G r 1 suc tat P r as no internal vertices in common wit G r 1. Te decomposition G r = x + P 1 + P 2 + +P r is called an (bipartite) ear decomposition of G r. It was sown in [13] tat a bipartite grap is elementary if and only if it as an (bipartite) ear decomposition. An ear decomposition (G 1,G 2,,G r (=G)) (equivalently, G = x + P 1 + P 2 + +P r ) of a plane elementary bipartite grap G is called a RFD if G 1 is te boundary of an interior face of G and te it ear P i lies in te exterior of G i 1 suc tat P i and a part of te peripery of G i 1 surround an interior face of G for all 2 i r. Teorem 1 (Zang and Zang [22]). Let G be a plane bipartite grap oter tan K 2. Ten G is elementary if and only if G as a RFD starting wit te boundary of any interior face of G. Teorem 1 gives te construction metod for plane elementary bipartite graps: starting wit some face, ten adding one new face at eac step gives any plane elementary bipartite grap. In tis paper we are interested in reversing tis procedure for elementary benzenoid graps. Namely, for a given elementary benzenoid grap we want to find a sequence of faces (exagons) tat decompose te grap in suc a manner tat te grap obtained at eac step of te decomposition is elementary. A face f of a plane bipartite grap G is periperal if te periperies of G and f ave a non-empty intersection. Let G be a plane bipartite grap. Let f be a periperal face of G and P a common pat of te periperies of f and G. Let G f denote te resultant subgrap of G by removing te internal vertices and edges of P. If G f is elementary tan we call f a reducible face of G. Te following teorem presented in [22] confirms te existence of reducible faces in plane elementary bipartite grap. Teorem 2. Let G be a plane elementary bipartite grap wit at least two finite faces. Ten G as at least two reducible faces. Trougout te paper, for a given grap G, let n stand for te number of its vertices.

3 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) In te next section we caracterize te reducible faces of an elementary benzenoid grap. Te caracterization is te basis for an algoritm wic finds a reducible face decomposition for a given grap of tis class. Tis algoritm wit te time complexity O(n 2 ) is presented in Section 3. In Section 4 we define pericondensed components of an elementary benzenoid grap and improve te running time of te algoritm for elementary benzenoid graps wit at most one pericondensed component. In particular, we prove tat every grap G of tis class contains a reducible exagon tat can be obtained on te basis of te so-called minimal 1-factor. Furtermore, we sow tat te minimal 1-factor of te grap obtained after te removal of from G can be computed in linear time. Tis result gives te linear algoritm to find a RFD for an elementary benzenoid grap wit at most one pericondensed component. 2. Caracterization of reducible faces Let M be a matcing of G. A vertex of G is called saturated by M if it is matced and unsaturated if it is not matced. A pat P is M-alternating if edges of P appear alternately in and off te M. If te endpoints of P are unsaturated, ten P is an augmenting pat. A cycle C is M-alternating if edges of C appear alternately in and off te M.AnM-alternating cycle C of G is said to be proper (improper) if every edge of C belonging to M goes from wite (black) end-vertex to black (wite) end-vertex by te clockwise orientation of C. A face f of G is said to be resonant if G as a 1-factor M suc tat te boundary of f is an M-alternating cycle. In [22], te following two teorems are proven (Teorem 4 is also obtained in [17]). Teorem 3. A non-trivial plane bipartite grap is elementary if and only if every face is resonant. Let us call te boundary of te infinite face of G te boundary or te outer cycle of G. Teorem 4. A benzenoid grap G is elementary if and only if te boundary of G is resonant. Let G be a plane bipartite grap. Let M(G) denote te set of all 1-factors of G. It was sown in [21] tat G as a unique 1-factor Mˆ0 suc tat G as no proper Mˆ0 -alternating cycles. We call Mˆ0 te minimal 1-factor of G, since Mˆ0 is te minimal element of te poset induced by M(G) [11,12]. In addition, G as a unique 1-factor Mˆ1 suc tat G as no improper Mˆ1 -alternating cycles. Mˆ1 is called te maximal 1-factor of G. A monotone pat system of a benzenoid grap G is a set of disjoint monotonically decreasing pats of G in wic eac pat issues at a peak and ends at a valley. A perfect pat system of G is a monotone pat system wic covers all peaks and valleys. It is sown in [15] tat a benzenoid grap as a 1-factor if and only if it as a perfect pat system. Moreover, it is proved in [5] tat if a benzenoid grap as a perfect pat system ten te induced matcing between peaks and valleys is unique. From a perfect pat system we construct te corresponding 1-factor by including: all non-vertical edges in te monotone pats and all te vertical edges not in te monotone pats. If G as a perfect pat system ten G admits a bijection between te set of peaks and te set of valleys. Tis fact induces te set B(G) suc tat (p, v) B(G) if and only if a peak p and a valley v are connected wit a monotone pat of a perfect pat system. Let P be a monotone pat from a peak to a valley in G. Ten te subgrap G P obtained by deleting all vertices of P and teir incident edges from G may ave more tan one component. A component of G P is said to be a left (rigt) component if te edges between P and te component itself are on te left (rigt) of P. Te left (rigt) bank of P is composed of all left (rigt) components of G P, P itself and all te edges between P and tese components. Let P p,v denote te monotone pat between a peak p and a valley v of a perfect pat system P. A perfect pat system L is said to be te leftmost perfect pat system of G if every monotone pat P between p and v is on te rigt bank of L p,v L. Analogously, a perfect pat system R is said to be te rigtmost perfect pat system of G if every monotone pat P between p and v is on te left bank of R p,v R.

4 1714 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) e 1 e 2 e 6 e 3 e 5 e 4 Fig. 2. A proper alternating face. Te symmetry difference of finite sets A and B is defined as A B :=(A B)\(A B).If is a exagon of a benzenoid grap G and M a 1-factor of G ten in te M operation, is always regarded as te set of edges bounding te exagon. Proposition 1. Let G be an elementary benzenoid grap. Te 1-factor induced by te rigtmost perfect pat system of G is te minimal 1-factor of G. Proof. Let M denote te 1-factor of G induced by its rigtmost perfect pat system R. Suppose tat in G exists a exagon suc tat te boundary of forms a proper M-alternating cycle. Denote te edges of as can be seen in Fig. 2. Since te edges e 1 and e 5 are non-vertical, tere exists a monotonically decreasing pat R p,v R suc tat te edges e 1 and e 5 belong to R p,v. Note tat te edges of {e 2,e 3,e 4 } do not intersect wit any oter monotonically decreasing pat of R. Terefore P :=R p,v is a monotonically decreasing pat between p and v in (R\R p,v ) P. However, P is not on te left bank of R p,v and we obtained a contradiction. Analogously we prove te following proposition. Proposition 2. Let G be an elementary benzenoid grap. Te 1-factor induced by te leftmost perfect pat system of G is te maximal 1-factor of G. Let G be a benzenoid grap. Let us define te set of edges W(G) ={R p,v L p,v ; (p, v) B(G)}. In oter words, W is te set of edges belonging to R p,v or L p,v but not to bot. Te definition stated above and te following result appeared in [4]. Proposition 3. Te edges of a maximal cycle C induced by W(G) togeter wit te edges of te interior of C compose te edges of te corresponding elementary component of G. An 1-factor M is said to be periperal if te outer cycle of G is M-alternating. Te next propositions sow tat te minimal and te maximal 1-factor of G are periperal. Proposition 4. Let G be an elementary benzenoid grap. Ten te outer cycle of G is improper Mˆ0 -alternating as well as proper Mˆ1 -alternating. Proof. Since G is elementary, from Proposition 3 it follows tat all te edges of te boundary of G belong to W(G). An edge of te outer cycle is terefore eiter in a pat of te rigtmost perfect pat system R p,v or in a pat of te leftmost perfect pat system L p,v, (p, v) B(G). We call an edge of te outer cycle left or rigt weter it belongs to L p,v or R p,v, respectively. Let M denote te set tat contains all non-vertical rigt edges and all vertical left edges. From Proposition 1 it follows tat M is a subset of te minimal 1-factor of G.

5 We first sow te following: A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) Claim 1. A left and a rigt edge of G are adjacent if and only if teir common vertex is eiter te peak or te valley of teir common exagon. Proof. Let p be a peak of te exagon tat joins edges e and f. Note tat teir common exagon does not need to be in te interior of G. Since a perfect pat system is composed of monotonically decreasing pats, e and f cannot be in te same pat. Moreover, since te pats of a perfect pat system are disjoint, e and f cannot belong to te pats of te same perfect pat system. It follows tat one of te edges as to be left and te oter rigt. If two edges are joined wit a valley, te proof goes analogously. Conversely, let edges e and f be joined by a vertex u tat is neiter a peak nor a valley. Suppose witout loss of generality tat e is above f. Note tat if e is vertical ten f is non-vertical and vice versa. Suppose first tat e is non-vertical and it belongs to a monotonically decreasing pat P p,v. Since te joint vertex of e and f is neiter te peak p nor te valley v, it follows tat f as to be also in P p,v. In oter words, e and f are eiter bot left or bot rigt. If e is vertical, assume tat f belongs to some monotonically decreasing pat. From te analogous arguments as above we can also conclude tat e and f are eiter bot left or bot rigt. To conclude te proof of te proposition we first sow tat te boundary of G is M-alternating (and terefore also Mˆ0 -alternating). From Claim 1 follows tat if te common vertex of two edges is te peak or te valley of teir common exagon, ten one of tese two edges is in and te oter off M. Moreover, since te left (rigt) edges of a leftmost (rigtmost) pat appear alternately in and off M, te assertion follows. To see te orientation of te edges of M observe te edges joined at a peak of G. From te arguments above follows, tat te rigt edge is in and te left edge off M. Moreover, te rigt edge goes from black end-vertex to wite end-vertex by te clockwise orientation of te boundary cycle. Tese conclusions complete te proof tat te outer cycle of G is improper Mˆ0 -alternating. To sow tat te outer cycle of G is proper Mˆ1 -alternating, denote wit M te set tat contains all vertical rigt edges and all non-vertical left edges wic is a subset of te maximal 1-factor of G. Te rest of te proof goes analogously as above. Teorem 5. Let G be an elementary benzenoid grap. Ten is a reducible exagon of G if and only if te following old: (i) te common peripery of and G is a pat of odd lengt and (ii) G admits a periperal 1-factor M suc tat te edges of form an M-alternating cycle. Proof. Let be a reducible exagon of G and let C denote te outer cycle of G. Ten te common peripery of and G is a pat P of lengt d = 1, 3, 5. Since is reducible, G is elementary. Let Mˆ0 and Mˆ1 denote te minimal and te maximal 1-factor of G, respectively. (i) d = 1. Let e denote te edge of P. e can be eiter in Mˆ1 or in Mˆ0. Bot cases are illustrated in Fig. 3a and b, respectively. If e is in Mˆ1, consider te minimal 1-factor of G denoted M. Clearly, = ˆ0 M ˆ0 Mˆ0 is also te minimal 1-factor of G. Moreover, Mˆ0 C is a periperal 1-factor in G and te edges of form an Mˆ0 C-alternating cycle. If e is in Mˆ0, consider te maximal 1-factor of G denoted. Ten = M ˆ1 M ˆ1 Mˆ1 is also te maximal 1-factor of G. Analogously, Mˆ1 C is periperal 1-factor in G and te edges of form an Mˆ1 C-alternating cycle. (ii) d =3. P can be eiter Mˆ1 -augmenting or Mˆ0 -augmenting. Bot cases are illustrated in Fig. 4a and b, respectively. If P is Mˆ1 -augmenting, consider te maximal 1-factor of G denoted. is not 1-factor of G and bot endvertices M ˆ1 M ˆ1 of exactly one edge of are unsaturated (see Fig. 4a). We call tis edge e. Clearly, M {e} is te maximal 1-factor ˆ1 of G. Moreover, (M {e}) C is a periperal 1-factor in G and te edges of form an {e}) C-alternating ˆ1 (M ˆ1 cycle. If P is Mˆ0 -augmenting, analogously as above consider te minimal 1-factor of G. (iii) d = 5. From Teorem 3 follows tat te edges of form an Mˆ0 -alternating cycle or an Mˆ1 -alternating cycle.

6 1716 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) e 1 e 1 e e e 2 e 2 Fig. 3. Cases were d = 1. e e Fig. 4. Cases were d = 3. For te converse suppose tat te common peripery of and G is a pat P of odd lengt. Suppose also tat G admits a periperal 1-factor M suc tat te edges of form an M-alternating cycle. Clearly, P is of lengt d = 1, 3, 5. Let C denote te outer cycle of G. (i) d = 1. Let e denote te edge of P. Te edges of form an M-alternating cycle, tus e as to be in M. Since M C is a periperal 1-factor in G, from Teorem 4 follows tat G is elementary and te case is settled. (ii) d = 3. Since M is periperal and te edges of form an M-alternating cycle, P cannot be augmenting. It follows tat tere is exactly one edge e of P tat does not belong to M. Clearly, e is in M C. Moreover (M C)\{e} is a periperal 1 -factor in G. Terefore, by Teorem 4, G is elementary and te assertion follows. (iii) d = 5. Let e denote te edge of wic is not in P. Clearly e is in M. Furtermore, te restriction of M to G is a periperal 1-factor in G. Analogously as above we can see tat G is elementary and terefore is reducible.

7 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) Te algoritm From Teorem 2 and te only if part of te proof of Teorem 5 we get te following: Corollary 1. Let G be an elementary benzenoid grap wit te outer cycle C. Ten G contains at least one reducible exagon, suc tat te edges of form Mˆ0 C-alternating or Mˆ1 C-alternating cycle. Let G be an elementary benzenoid grap wit te RFD (G 1,G 2,,G r (=G)). Te sequence of exagons 1, 2, 3,, r is called reducible if i is a reducible face of G i suc tat G i 1 = G i i, i = r, r 1,,2 and 1 = G 1. Teorem 5 caracterizes te reducible exagons of an elementary benzenoid grap. Moreover, in conjunction wit Corollary 1 it gives te basis for te following algoritm wic finds a reducible sequence of exagons. Algoritm RFD input: an elementary benzenoid grap G. output : a reducible sequence of exagons L i, i = 1,,r. i :=1. repeat 1. Mˆ0 := te minimal 1-factor of G. 2. Mˆ1 := te maximal 1-factor of G. 3. C := te boundary cycle of G. 4. L i := a periperal exagon suc tat te edges of form an Mˆ0 C-alternating or Mˆ1 C-alternating cycle. 5. G := G. 6. i := i + 1. until G is a single exagon. L i :=. Teorem 6. Algoritm RFD finds a RFD of an elementary benzenoid grap G and can be implemented to run in O(n 2 ) time. Proof. Te correctness of te algoritm follows by Teorem 5 and Corollary 1. Starting from G = G r, te algoritm at eac execution of te loop finds a reducible exagon in G = G i and ten removes tis exagon from G i. Te obtained grap G i 1 is elementary, terefore we can repeat te procedure till te last exagon. Concerning te time complexity of te algoritm, we first sow tat te body of te loop is executed in linear time. Note tat a vertex of G possesses at most tree adjacent vertices. Tus, te complexities of basic operations: deleting an edge, deleting a vertex, deleting all edges incident wit a vertex, etc., are constant notwitstanding a representation of G. For Steps 1 and 2 we invoke routines RPS and LPS presented in [5] wic compute te rigtmost perfect pat system and te leftmost perfect pat system of G in linear time. Propositions 1 and 2 ten imply tat Steps 1 and 2 of RFD can be executed witin te same time bound. Te boundary cycle of G can be clearly computed in linear time by traversing te edges of G. If we mark te edges belonging to C, ten we can obtain Mˆ0 C and Mˆ1 C in a time wic is linear in te number of edges of C. For Step 4 furter observe tat G clearly admits less tan n periperal exagons. Moreover, we can ceck in constant time weter a periperal exagon induces a Mˆ0 C-alternating or Mˆ1 C-alternating cycle. Since te degree of a vertex in G is constant, tis time bound also olds for Step 5. Te time complexity of reducing one exagon is terefore linear in te number of edges of G. Finally, since te loop executes O(n) times, it follows tat te overall time complexity of te algoritm is O(n 2 ). Remark. Fig. 5 sows an example of an elementary benzenoid grap G wit te minimal 1-factor Mˆ0 and te maximal 1-factor Mˆ1. Let C denote te boundary cycle of G. Since te exagon denoted induces a Mˆ1 C-alternating cycle, by Teorem 5 is reducible.

8 1718 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) G wit te minimal 1-factor G wit te maximal 1-factor e e Fig. 5. Te minimal (left) and te maximal (rigt) 1-factor of G. G- wit te minimal 1-factor G- wit te maximal 1-factor Fig. 6. Te minimal (left) and te maximal (rigt) 1-factor of G. Note tat te maximal 1-factor of G can be obtained simply by removing te edge e from Mˆ1. However, in order to obtain te minimal 1-factor of G from Mˆ0, one sould remove and replace O(n) edges as can be seen in Fig. 6. Tus, te example clearly sows te necessity of recalculating te minimal and te maximal 1-factor of te grap after a reduction. 4. Pericondensed components Let G be a plane grap. Te vertices of te inner dual of G are te finite faces of G, two vertices being adjacent if and only if te corresponding faces sare an edge in G. Te inner dual of a benzenoid grap is a subgrap of te regular triangular grid (see Fig. 7). Clearly, te inner dual of a catacondensed benzenoid grap is a tree wit maximum vertex degree tree. A subgrap H of G is a block of G if H is a maximal subgrap witout cut vertices or edges wose removal increases te number of components of G. Let G be a benzenoid grap. Te subgrap of G tat corresponds to te block of te inner dual of G is called a pericondensed component of G. Te subgrap of G obtained by removing te vertices and te edges of all pericondensed components of G we call a catacondensed forest of G, wile its component is called a catacondensed tree. A catacondensed tree is called a link if it joins te vertices of two pericondensed components and a beam oterwise. Tese definitions are illustrated in Fig. 7 wit encircled components of te grap s catacondensed forest. Let G be an elementary benzenoid grap and C te outer cycle of G. A exagon is easily reducible if is reducible and te minimal 1-factor of G can be obtained from te minimal 1-factor of G in constant time. Te remark of te previous section sows tat a reducible exagon need not to be easily reducible. Note tat a exagon of a benzenoid grap G is periperal if te periperies of G and ave a non-empty intersection. Te peripery of contains one, two, tree, four or five edges. Wit respect to tis, we say tat a periperal exagon is of type T1, T2, T3, T4 or T5. We will need te following simple lemma. Lemma 1. Let G be an elementary benzenoid grap. Ten every exagon of G of type T5 is easily reducible. Proof. Te proof of Teorem 5 implies tat every exagon of G of type T5 is reducible. Let Mˆ0 denote te minimal 1-factor of G and a exagon of type T5. Furtermore, let e denote te joint edge of G and. Ten we can obtain

9 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) link beam Fig. 7. A benzenoid grap wit its inner dual and catacondensed forest. te minimal 1-factor of G by removing te edges of te peripery of from Mˆ0 and, if e/ Mˆ0, by including e. It is straigtforward to see tat tis operation can be performed in constant time and te assertion follows. We will sow tat a RFD can be obtained in linear time for elementary benzenoid graps containing at most one pericondensed component. Te algoritm is based on te following teorem. Teorem 7. Let G be an elementary benzenoid grap containing at most one pericondensed component. Ten G contains at least one easily reducible exagon. From Lemma 1 it follows tat it suffices to prove te teorem for a pericondensed component of G. Note tat all exagons of a pericondensed component are of type T1, T2, T3 or T4. Te proof of te teorem is based on te claims presented below. In tese claims we adopt te following conventions: G is an elementary benzenoid grap wit exactly one pericondensed component and witout beams. P u,v is a pat of te rigtmost perfect pat system R suc tat no oter pat of R is in te left bank of P u,v. Te edges of a exagon are denoted as in Fig. 2. Claim 2. Let be a exagon of G in te left bank of P u,v suc tat is of type T1 or type T2 and P u,v =. Ten as no adjacent exagon on te left-and side of its vertical edge. Proof. Let be a exagon of G in te left bank of P u,v suc tat P u,v =.If is of type T1 ten cases (b), (c), (e) and (f) in Fig. 8 cannot exist since tey imply a new peak or valley in te left bank of P u,v. Te case (d) in Fig. 8 cannot occur since would clearly not lie in te left bank of P u,v. In order to consider a exagon of type T2, note tat since G is pericondensed, te exagons adjacent to ave to remain connected in G. Some possible cases are depicted in Fig. 9. Two cases not depicted in te figure are in contradiction wit te assumption tat is in te left bank and as no intersection wit P u,v. Te cases (c) and (d) cannot exist in te left bank of P u,v since tey imply a new peak or valley, respectively. Let e be an edge of a benzenoid grap G. Ten te cut C e corresponding to e is te set of edges so tat wit every edge e of C e also te opposite edge wit respect to a exagon containing e belongs to C e. (As benzenoid graps admits

10 1720 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) Fig. 8. Possible cases were a type T1 exagon is in te left bank of P u,v. Fig. 9. Possible cases were a type T2 exagon is in te left bank of P u,v. isometric embeddings into ypercubes [7], C e can also be described as te equivalence class of te Djoković Winkler relation Θ containing e, cf. [6].) Claim 3. Let be a exagon of G in te left bank of P u,v suc tat is of type T1 or type T2 and P u,v =. Ten in te left bank of P u,v exists a exagon of type T3 wic as an empty intersection wit P u,v. Proof. Suppose is of type T1. We now consider te cut C e4, were te edge e 4 corresponds to. Te situation is depicted in Fig. 10a. Let be te exagon containing te last edge e in te cut looking from e 4 towards e 1. Since P u,v is monotonous, P u,v =. In oter words, te peak p is on te rigt-and side of te cut looking in te described direction. Since is te last exagon of te cut, e 1 as to be on te boundary G. Suppose is of type: T1: Tis implies tat only te edge e 1 of is on te boundary G. But by Claim 2 tis is a contradiction. T2: Tis implies tat besides e 1 of, since G is pericondensed, eiter e 2 or e 6 is on te peripery of G. Ife 2 is on te boundary of G, ten by Claim 2 tis is a contradiction. If e 6 is on te boundary of G, ten G would admit a new valley on te left bank of P u,v and we again obtain a contradiction. T4: Ten contains te peak of G wic leads to a contradiction.

11 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) peak u e 2 e 5 e 1 e 4 valley v Fig. 10. Te exagons of te cut. Similar argument can be used if is of type T2. If te case from Fig. 9b occurs, we again observe te last exagon in te cut C e4. Te situation is analogous as depicted in Fig. 10a wit te exception tat te exagon is of type T2. If te case from Fig. 9a occurs, te situation is depicted in Fig. 10b. We now consider te cut C e2 and te exagon on te left-and side of te valley v. We ave proved tat is of type T3 and te proof is complete. Let p and v denote te exagon wic contains te peak p and te valley v, respectively. Claim 4. Let every exagon in te left bank of P u,v ave a non-empty intersection wit P u,v and let all of tese exagons different from p and v be of type T2. Ten te exagon containing te peak p and/or te exagon containing te valley v are of type T3. Proof. Clearly, p and v are eiter of type T3 or T4. Suppose now tat p and v are bot of type T4. Since p, v and te exagons between tem ave a non-empty intersection wit P u,v, we distinguis two cases illustrated in Fig. 11. We first consider te case from Fig. 11a. Ten v as two adjacent exagons, wit one it sares te edge e 1 and wit te oter te edge e 2. Denote tem wit and, respectively. Note tat e 4 of is on te boundary of G, moreover it is in Mˆ0 and since it is non-vertical, it belongs to P u,v. But ten as an empty intersection wit P u,v and we obtained a contradiction. It follows tat v as to be of type T3. In order to prove te case from Fig. 11b observe te exagons adjacent to p. Te rest of te proof goes analogously as above. Claim 5. Let every exagon in te left bank of P u,v ave a non-empty intersection wit P u,v and let bot p and v be of type T4. Ten at least one of te exagons on te left bank of P u,v is of type T1. Proof. From Claim 4 it follows tat at least one of te exagons between p and v cannot be of type T2. Let us denote it by. Note first tat cannot be of type T4, since it would imply a new peak in te left bank of P u,v. Suppose ten tat is of type T3. Te situation is depicted in Fig. 12b. By assumption, intersects wit P u,v. Now consider te exagons adjacent to p. Let us denote tem as and. G is elementary, tus from Proposition 4 it follows tat te edge e 2 of as to be in te minimal 1-factor of G. Since P u,v is monotonous, te edge e 1 of is also in te minimal 1-factor of G. But edges e 1 and e 2 are incident and we obtained a contradiction. We sowed tat as to be of type T1 (te situation is depicted on Fig. 12a) and te assertion follows.

12 1722 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) p v v Fig. 11. Te peak and valley of type T4. p v Fig. 12. Hexagon of type T1. Proof of Teorem 7. If G is catacondensed or it as at least one beam, ten from Lemma 1 te assertion clearly follows. Suppose ten tat G is an elementary benzenoid grap wit exactly one pericondensed component and witout beams. Let C denote te outer cycle of G and let Mˆ0 and Mˆ1 denote te minimal and te maximal 1-factor of G, respectively. Let P u,v be a pat of te rigtmost perfect pat system R of G suc tat none of te oter pats of R is in te left bank of P u,v. We will sow tat at least one easily reducible exagon exists in te left bank of P u,v.

13 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) p Fig. 13. Case were p is of type T3. We first assume tat in te left bank of P u,v exists a exagon suc tat as no intersection wit P u,v. We claim tat in te left bank of P u,v exists a exagon, suc tat is of type T3 and P u,v =. Note first tat cannot be of type T4, since it would imply a new peak in te left bank of P u,v.if is of type T3, ten we state := and we are done. Oterwise, is of type T1 or T2 and by Claim 3 te requested exagon also exists. Note now tat if is disjoint wit P u,v, ten by Proposition 1 bot vertical edges of are in te minimal 1-factor of G (see te exagon in Fig. 10). But ten te edges of form an Mˆ0 C-alternating cycle and from Teorem 5 it follows tat is reducible. Furtermore, te minimal 1-factor of G can be obtained by removing te vertical edge of on te outer cycle from te minimal 1-factor of G. Since tis can clearly be done in constant time, is easily reducible. Suppose now tat every exagon in te left bank of P u,v as at least one edge in common wit P u,v. Claims 4 and 5 sow tat in te left bank of P u,v exists eiter a exagon of type T3 (at te peak p or at te valley v) or a exagon of type T1. Let denote tis exagon. Suppose is of type T3 and suppose it contains te peak p (see Fig. 13). Note tat P p,v is a pat of te rigtmost perfect pat system. It follows tat is Mˆ0 -resonant and tus reducible. Te minimal 1-factor of G can be M ˆ0 obtained from te minimal 1-factor Mˆ0 of G by removing te edges e 2,e 4 and e 6 of from Mˆ0 and by adding te edges e 3 and e 5 of. Tis can clearly be done in constant time, tus is easily reducible. Finally, if is of type T1 (te situation is depicted in Fig. 12a), ten te edges e 2 and e 4 of belong to P u,v. Moreover, te edge e 6 of is vertical, terefore all of tese tree edges belong to Mˆ0. It follows tat is Mˆ0 -resonant and terefore reducible. Te minimal 1-factor M of G can be obtained from te minimal 1-factor ˆ0 Mˆ0 of G by removing edges e 2,e 4 and e 6 from Mˆ0 and by adding te edges e 1,e 3 and e 5. Again, tis procedure requires constant time, wic yields tat is easily reducible and te proof is complete. Te following algoritm finds a reducible sequence of exagons of an elementary benzenoid grap wit at most one pericondensed component in linear time. Algoritm RFD-PC input : an elementary benzenoid grap G wit at most one pericondensed component. output : a reducible sequence of exagons L i, i = 1,,r. 1. Mˆ0 : = te minimal 1-factor of G. 2. H : = te set of all easily reducible exagons of type T1, T3 and T5. 3. i :=1. 4. wile H = do (a) Remove an arbitrary exagon from H. (b) Remove from G. (c) Update Mˆ0. (d) Find all neigbors of and put tem in H if necessary. (e) L i :=. (f) i :=i Add te remaining exagon to RFD.

14 1724 A. Taranenko, A. Vesel / Discrete Applied Matematics 156 (2008) Teorem 8. Algoritm RFD-PC finds a RFD of a given elementary benzenoid grap wit at most one pericondensed component in linear time. Proof. Correctness of te algoritm follows from Teorems 5 and 7. Note tat te latter imply tat after te removal of an easily reducible exagon te minimal 1-factor of te obtained grap differs from old Mˆ0 only in edges wic are in te intersection wit or adjacent to. It follows tat a new easily reducible exagon can appear only in te neigborood of. Concerning te complexity of te algoritm, we again invoke te procedure RPS presented in [5] wic computes te rigtmost perfect pat system in linear time. Tis implies tat Step 1 of RFD-PC can be executed witin te same time bound. By Teorem 7 tere exists at least one easily reducible exagon. All easily reducible exagons are located on te boundary of G and we can detect tem in linear time. Te number of executions of wile loop of te algoritm is bounded wit te number of all exagons in G. It remains to be proven tat all steps witin te body of te loop can be executed in constant time. Hexagons in H are all easily reducible, terefore te removal of tese exagons from G and finding te new minimal 1-factor can be done in constant time. Furtermore, every reducible exagon as at most five adjacent exagons, terefore te computation of Step 4(d) is independent of n. Tis concludes our proof. Since te complexity of RFD-PC is proportional to te size of te input, te algoritm is optimal in a precise sense wic is in common use in te teory of computational complexity. However, it also raises te natural question, weter te algoritm wit te best possible complexity can be obtained for more general class of elementary benzenoid graps. References [1] R. Cen, F. Zang, Hamilton pats in Z-transformation graps of perfect matcings of exagonal systems, Discrete Appl. Mat. 74 (1997) [2] S.J. Cyvin, I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons, Springer, Heidelberg, [3] I. Gutman, S.J. Cyvin, Introduction to te Teory of Benzenoid Hydrocarbons, Springer, Berlin, [4] P. Hansen, M. Zeng, A linear algoritm for fixed bonds in exagonal systems, J. Mol. Struct. (Teocem) 257 (1992) [5] P. Hansen, M. Zeng, A linear algoritm for perfect matcing in exagonal systems, Discrete Mat. 122 (1993) [6] W. Imric, S. Klavžar, Product Graps: Structure and Recognition, Wiley, New York, [7] S. Klavžar, I. Gutman, B. Moar, Labeling of benzenoid systems wic reflects te vertex-distance relations, J. Cem. Inf. Comput. Sci. 35 (1995) [8] S. Klavžar, A. Vesel, P. Žigert, On resonance graps of catacondensed exagonal graps: structure, coding, and Hamilton pat algoritm, MATCH Commun. Mat. Comput. Cem. 49 (2003) [9] S. Klavžar, A. Vesel, P. Žigert, I. Gutman, Binary coding of Kekulé structures of catacondensed benzenoid ydrocarbons, Comput. & Cem. 25 (2001) [10] S. Klavžar, P. Žigert, G. Brinkmann, Resonance graps of catacondensed even ring systems are median, Discrete Mat. 253 (2002) [11] P.C.B. Lam, W.C. Siu, H. Zang, Resonance graps and a binary coding for te 1-factors of benzenoid systems, manuscript. [12] P.C.B. Lam, H. Zang, A distributive lattice on te set of perfect matcings of a plane biparite grap, Order 20 (2003) [13] L. Lovász, M.D. Plummer, On minimal elementary bipartite graps, J. Combin. Teory Ser. B 23 (1977) [14] L. Lovász, M.D. Plummer, Matcing Teory, Nort-Holland, Amsterdam, [15] H. Sacs, Perfect matcings in exagonal systems, Combinatorica 4 (1) (1980) [16] A. Vesel, Caracterization of resonance graps of catacondensed exagonal graps, MATCH Commun. Mat. Comput. Cem. 53 (2005) [17] F. Zang, R. Cen, Wen eac exagon of a exagonal system covers it, Discrete Appl. Mat. 30 (1991) [18] F. Zang, X. Guo, Te enumeration of several classes of exagonal systems, Acta Mat. Appl. Sinica (Englis Ser.) 15 (1999) [19] F. Zang, X. Guo, R. Cen, Z-transformation graps of perfect matcings of exagonal systems, Discrete Mat. 72 (1988) [20] F. Zang, H. Zang, A note on number of perfect matcings of bipartite graps, Discrete Appl. Mat. 73 (1997) [21] H. Zang, F. Zang, Te rotation graps of perfect matcings of plane bipartite graps, Discrete Appl. Mat. 73 (1997) [22] H. Zang, F. Zang, Plane elementary bipartite graps, Discrete Appl. Mat. 105 (2000)