Characterization of a class of graphs related to pairs of disjoint matchings

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  • Discrete Mathematics 309 (2009)

    Characterization of a class of graphs related to pairs ofdisjoint matchingsI

    Anush Tserunyan

    Department of Informatics and Applied Mathematics, Yerevan State University, Yerevan, 0025, Armenia

    Received 14 December 2006; received in revised form 31 December 2007; accepted 4 January 2008Available online 10 March 2008

    Dedicated to my mother, father and sister Arevik


    For a given graph consider a pair of disjoint matchings the union of which contains as many edges as possible. Furthermore,consider the ratio of the cardinalities of a maximum matching and the largest matching in those pairs. It is known that for anygraph 54 is the tight upper bound for this ratio. We characterize the class of graphs for which it is precisely

    54 . Our characterization

    implies that these graphs contain a spanning subgraph, every connected component of which is the minimal graph of this class.c 2008 Elsevier B.V. All rights reserved.

    Keywords: Matching; Pair of disjoint matchings; Maximum matching

    1. Introduction

    In this paper we consider finite undirected graphs without multiple edges, loops, or isolated vertices. Let V (G) andE(G) be the sets of vertices and edges of a graph G, respectively. By (G) we denote the cardinality of a maximummatching of G.

    Let B2(G) be the set of pairs of disjoint matchings of a graph G. Set:

    (G).= max{|H | + |H | : (H, H ) B2(G)}.

    Furthermore, let us introduce another parameter:

    (G).= max{|H |, |H | : (H, H ) B2(G) and |H | + |H | = (G)},

    and define a set:

    M2(G).= {(H, H ) B2(G) : |H | + |H | = (G) and |H | = (G)}.

    While working on the problems of constructing a maximum matching F of a graph G such that (G \ F) ismaximized or minimized, Kamalian and Mkrtchyan designed polynomial algorithms for solving these problems for

    I The current work is the main part of the authors Masters thesis defended in May 2007. It was supported by a grant of Armenian NationalScience and Education Fund.

    E-mail address:

    0012-365X/$ - see front matter c 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.disc.2008.01.004

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    trees [4]. Unfortunately, the problems turned out to be NP-hard already for connected bipartite graphs with maximumdegree three [5], thus there is no hope for the polynomial time calculation of 1(G) even for bipartite graphs G, where

    1(G) = max{(G \ F) : F is a maximum matching of G}.Note that for any graph G

    (G) = (G)+ 1(G) if and only if (G) = (G).Thus, 1(G) can be efficiently calculated for bipartite graphs G with (G) = (G) since (G) can be calculatedfor bipartite graphs by using a standard algorithm of finding a maximum flow in a network. Let us also note that thecalculation of (G) is NP-hard even for the class of cubic graphs since the chromatic class of a cubic graph G is threeif and only if (G) = |V (G)| (see [3]).

    Being interested in the classification of graphs G, for which (G) = (G), Mkrtchyan in [7] proved a sufficientcondition, which due to [1,2], can be formulated as: if G is a matching covered tree then (G) = (G). Note thata graph is said to be matching covered (see [8]) if its every edge belongs to a maximum matching (not necessarily aperfect matching as it is usually defined, see e.g. [6]). Very deep results and interesting conjectures involving theseparameters will be given in [10].

    In contrast with the theory of 2-matchings, where every graph G admits a maximum 2-matching that includes amaximum matching [6], there are graphs (even trees) that do not have a maximum pair of disjoint matchings (a pairfrom M2(G)) that includes a maximum matching.

    The following is the best result that can be stated about the ratio (G)(G) for any graph G (see [9]):

    1 (G)(G)


    The aim of the paper is the characterization of the class of graphs G, for which the ratio (G)(G) obtains its upper

    bound, i.e. the equality (G)(G) = 54 holds.

    Fig. 1. Spanner.

    Our characterization theorem is formulated in terms of a special graph called spanner (Fig. 1), which is the minimalgraph for which 6= (what is remarkable is that the equality

    = 54 also holds for spanner). This kind of theorem

    is common in graph theory: see [1] for characterization of planar or line graphs. Another example may be TuttesConjecture (now a beautiful theorem thanks to Robertson, Sanders, Seymour and Tomas) about the chromatic indexof bridgeless cubic graphs, which do not contain Petersen graph as a minor.

    On the other hand, let us note that in contrast with the examples given above, our theorem does not provide aforbidden/excluded graph characterization. Quite the contrary, the theorem implies that every graph satisfying thementioned equality admits a spanning subgraph every connected component of which is a spanner.

    2. Main notations and definitions

    Let G be a graph and dG(v) be the degree of a vertex v of G.

    Definition 2.1. A subset of E(G) is called a matching if it does not contain adjacent edges.

    Definition 2.2. A matching of G with maximum number of edges is called maximum.

    Definition 2.3. A vertex v of G is covered (missed) by a matching H of G, if H contains (does not contain) an edgeincident to v.

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    Definition 2.4. A sequence v0, e1, v1, . . . , vn1, en, vn is called a trail in G if vi V (G), e j E(G), e j =(v j1, v j ), and e j 6= ek if j 6= k, for 0 i n, 1 j, k n.

    The number of edges, n, is called the length of a trail v0, e1, v1, . . . , vn1, en, vn . A trail is called even (odd) if itslength is even (odd).

    Trails v0, e1, v1, . . . , vn1, en, vn and vn, en, vn1, . . . , v1, e1, v0 are considered equal. Trail T is also consideredas a subgraph of G, and thus, V (T ) and E(T ) are used to denote the sets of vertices and edges of T , respectively.

    Definition 2.5. A trail v0, e1, v1, . . . , vn1, en, vn is called a cycle if v0 = vn .Similarly, cycles v0, e1, v1, . . . , vn1, en, v0 and vi , ei+1, . . . , en, v0, e1, . . . , ei , vi are considered equal for any

    0 i n 1.If T : v0, e1, v1, . . . , vn1, en, vn is a trail that is not a cycle then v0, vn and e1, en are called the end-vertices and

    end-edges of T , respectively.

    Definition 2.6. A trail v0, e1, v1, . . . , vn1, en, vn is called a path if vi 6= v j for 0 i < j n.

    Definition 2.7. A cycle v0, e1, v1, . . . , vn1, en, v0 is called simple if v0, e1, v1, . . . , vn2, en1, vn1 is a path.

    Below we omit vi -s and write e1, e2, . . . , en instead of v0, e1, v1, . . . , vn1, en, vn when denoting a trail.

    Definition 2.8. For a trail T : e1, e2, . . . , en of G and i 1, define sets Ebi (T ), Eei (T ), Ei (T ) and V0(T ), V bi (T ),V ei (T ), Vi (T ) as follows:

    Ebi (T ).= {e j : 1 j min{n, i}},

    Eei (T ).= {e j : max{1, n i + 1} j n},

    Ei (T ).= Ebi (T ) Eei (T ),


    V0(T ).= {v V (G) : v is an end-vertex of T },

    V bi (T ).= {v V (G) : v is incident to an edge from Ebi (T )},

    V ei (T ).= {v V (G) : v is incident to an edge from Eei (T )},

    Vi (T ).= V bi (T ) V ei (T ).

    The same notations are used for sets of trails. For example, for a set of trails D, V0(D) denotes the set of end-vertices of all trails from D, that is:


    TDV0(T ).

    Let A and B be sets of edges of G.

    Definition 2.9. A trail e1, e2, . . . , en is called AB alternating if the edges with odd indices belong to A\B and othersto B \ A, or vice versa.

    If X is an AB alternating trail then X A (X B) denotes the graph induced by the set of edges of X that belong to A(B).

    The set of AB alternating trails of G that are not cycles is denoted by T (A, B). The subsets of T (A, B) containingonly even and odd trails are denoted by Te(A, B) and To(A, B), respectively. We use the notation C instead of T todenote the corresponding sets of AB alternating cycles (e.g. Ce(A, B) is the set of AB alternating even cycles).

    The set of the trails from To(A, B) starting with an edge from A (B) is denoted by T Ao (A, B) (TB

    o (A, B)).Now, let A and B be matchings of G (not necessarily disjoint). Note that AB alternating trail is either a path, or

    an even simple cycle.

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    Definition 2.10. An AB alternating path P is called maximal if there is no other AB alternating trail (a path or aneven simple cycle) that contains P as a proper subtrail.

    We use the notation MP instead of T to denote the corresponding sets of maximal AB alternating paths (e.g.MPBo (A, B) is the subset of MP(A, B) containing only those maximal AB alternating paths whose length is odd andwhich start (and also end) with an edge from B).

    Terms and concepts that we do not define can be found in [1,6,11].

    3. General properties and structural lemmas

    Let G be a graph, and A and B be (not necessarily disjoint) matchings of it. The following are properties of ABalternating cycles and maximal paths.

    First note that all cycles from Ce(A, B) are simple as A and B are matchings.

    Property 3.1. If the connected components of G are paths or even simple cycles, and (H, H ) M2(G), thenH H = E(G).Property 3.2. If C Ce(A, B) and v V (C) then dCA (v) = dCB (v).Property 3.3. Every edge e A4B1 lies either on a cycle from Ce(A, B) or on a path from MP(A, B).Property 3.4. (1) if F Ce(A, B) Te(A, B) then A and B have the same number of edges that lie on F,(2) if T T Ao (A, B) then the number of edges from A lying on T is one more than the number of ones from B.

    These observations imply:

    Property 3.5. |A| |B| = |MPAo (A, B)| |MPBo (A, B)|.Berges well-known theorem states that a matching M of a graph G is maximum if and only if G does not contain

    an M-augmenting path [1,6,11]. This theorem immediately implies:

    Property 3.6. If M is a maximum matching and H is a matching of a graph G then

    MPHo (M, H) = ,and therefore, |M | |H | = |MPMo (M, H)|.

    The proof of the following property is similar to the one of Property 3.6:

    Property 3.7. If (H, H ) M2(G) then MPH o (H, H ) = .Property 3.8. If (G) = 2(G) and (H, H ) B2(G) for which |H | + |H | = (G), then (H, H ), (H , H) M2(G) and MPo(H, H ) = .Proof. Let (H, H ) B2(G) such that |H | + |H | = (G). Then, max{|H |, |H |} (G)2 = (G). On the otherhand, by the definition of (G), max{|H |, |H |} (G). Thus, |H | = |H | = (G) and hence (H, H ) M2(G) aswell as (H , H) M2(G).

    Now we apply Property 3.7 to the pairs (H, H ) and (H , H), and get MPH o (H, H ) = and MPHo (H , H) = ,i.e. MPo(H, H ) = .

    Now let M be a fixed maximum matching of G. Over all (H, H ) M2(G), consider the pairs ((H, H ),M) forwhich |M (H H )| is maximized. Denote the set of those pairs by M2(G,M):

    M2(G,M).= {(H, H ) M2(G) : |M (H H )| is maximum}.

    Let (H, H ) be an arbitrarily chosen pair from M2(G,M).

    1 A4B denotes the symmetric difference of A and B, i.e. A4B = (A \ B) (B \ A).

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    Lemma 3.9. For every path P : m1, h1, . . . ,ml1, hl1,ml from MPMo (M, H)(1) m1,ml H ;(2) l 3.Proof. Let us show that m1,ml H . If l = 1 then P = m1, m1 M \ H , and m1 is not adjacent to an edge fromH as P is maximal. Thus, m1 H as otherwise we could enlarge H by adding m1 to it which would contradict(H, H ) M2(G). Thus, suppose that l 2. Let us show that m1 H . If m1 6 H then define

    H1.= (H \ {h1}) {m1}.

    Clearly, H1 is a matching, and H1 H = , |H1| = |H |, which means that (H1, H ) M2(G). But|M (H1 H )| > |M (H H )|, which contradicts (H, H ) M2(G,M). Thus m1 H . Similarly, it canbe shown that ml H .

    Now let us show that l 3. Due to Property 3.7, MPH o (H, H ) = , thus there is i, 1 i l, such thatmi M \ (H H ), since {m1,ml} H , and we have l 3. Lemma 3.10. Each vertex lying on a path from MPMo (M, H) is incident to an edge from H

    .Proof. Assume the contrary, and let v be a vertex lying on a path P from MPMo (M, H), which is not incident to anedge from H . Clearly, v is incident to an edge e = (u, v) M \ (H H ) lying on P .

    If u is not incident to an edge from H too, then H and H {e} are disjoint matchings and |H | + |H {e}| >|H | + |H | = 2(G), which contradicts (H, H ) M2(G).

    On the other hand, if u is incident to an edge f H , then consider the pair (H, H ), where H .= (H \{ f }){e}.Note that H and H are disjoint matchings and |H | = |H |, which means that (H, H ) M2(G). But|M (H H )| > |M (H H )| contradicting (H, H ) M2(G,M).

    For a path P MPMo (M, H), consider one of its end-edges f E1(P). Due to statement (1) of Lemma 3.9,f M H . By maximality of P , f is adjacent to only one edge from H , thus it is an end-edge of a path P f fromMPe(H, H )MPH o (H, H ). Moreover, P f MPe(H, H ) according to Property 3.7. Define a set Y MPe(H, H )as follows:

    Y (M, H, H ) .= {P f : P MPMo (M, H), f E1(P)}.

    Lemma 3.11.(1) The end-edges of paths of MPMo (M, H) lie on different paths of Y (M, H, H

    );(2) |Y (M, H, H )| = 2|MPMo (M, H)| = 2((G) (G)).(3) For every P Y (M, H, H ), P : h1, h1, . . . , hn, hn , where hi H , hi H, 1 i n,

    (a) h1 and h1 lie on a path from MPMo (M, H), but hn and hn do not lie on any path from MPMo (M, H);(b) n 2.

    Proof. (1) is true as otherwise we would have a path from Y (M, H, H ) with both end-edges from H contradictingY (M, H, H ) MPe(H, H ). Furthermore, (1) together with Property 3.6 imply (2).

    Now, let us prove (3a).By the definition of Y (M, H, H ), h1 is an end-edge of a path P1 from MPMo (M, H), and therefore h1 lies on P1 does not lie on any path from MPMo (M, H) as otherwise, due to Lemma 3.10, both vertices incident to hn would

    be incident to edges from H , which contradicts the maximality of P . Note that hn is not incident to an inner vertex(not an end-vertex) of a path P1 from MPMo (M, H) as any such vertex is incident to an edge from H lying on P1,and therefore different from hn . hn is incident neither to an end-vertex of a path P1 from MPMo (M, H) as it wouldcontradict the maximality of P1. Thus, hn is not adjacent to an edge lying on a path from MPMo (M, H), and thereforehn does not lie on any path from MPMo (M, H). The proof of (3a) is complete.

    Statement (3b) immediately follows from (3a). In particular, the statements (2) and (3b) of Lemma 3.11 imply |H | |H | 2|Y (M, H, H )| = 4((G)(G)).

    Taking into account that |H | = (G) and |H | = (G) (G), we get the following result (also obtained in [9]):Corollary 3.12. (G) (G) (G) 4((G) (G)), thus (G)

    (G) 54 .

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    4. Spanner, S-Forest and S-Graph

    The graph on Fig. 1 is called spanner. A vertex v of spanner S is called i-vertex, 1 i 3, if dS(v) = i . The3-vertex closest to a vertex v of spanner is referred as the base of v. The two paths of the spanner of length fourconnecting 1-vertices are called sides.

    For spanner S define sets U (S) and L(S) as follows:

    U (S).= {e E(S) : e is incident to a 1-vertex},

    L(S).= {e E(S) \U (S) : e is incident to a 2-vertex}.

    Note that for spanner S, and for every (H, H ) M2(S), the edge connecting the 3-vertices does not belong toH H ,...