Unicycle graphs and uniquely restrictedmaximum matchings

Vadim E. Levit 1

Department of Computer ScienceHolon Academic Institute of Technology

Holon, Israel

Eugen Mandrescu 2

Department of Computer ScienceHolon Academic Institute of Technology

Holon, Israel

Abstract

A matching M is called uniquely restricted in a graph G if it is the unique perfectmatching of the subgraph induced by the vertices that M saturates. G is a unicyclegraph if it owns only one cycle. Golumbic, Hirst and Lewenstein observed that fora tree or a graph with only odd cycles the size of a maximum uniquely restrictedmatching is equal to the matching number of the graph. In this paper we charac-terize unicycle graphs enjoying this equality. Moreover, we describe unicycle graphswith only uniquely restricted maximum matchings. Using these findings, we showthat unicycle graphs having only uniquely restricted maximum matchings can berecognized in polynomial time.

Keywords: uniquely restricted matching, local maximum stable set, greedoid.

1 Email: levitv@hait.ac.il2 Email: eugen m@hait.ac.il

Electronic Notes in Discrete Mathematics 22 (2005) 261–265

1571-0653/$ – see front matter © 2005 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2005.06.055

1 Introduction

Throughout this paper G = (V, E) is a simple (i.e., a finite, undirected, loop-less and without multiple edges) graph with vertex set V = V (G) and edgeset E = E(G).

A graph is unicycle if it owns only one cycle.

A stable set in G is a set of pairwise non-adjacent vertices. A stable setof maximum size will be referred to as a maximum stable set of G, and thestability number of G, denoted by α(G), is the cardinality of a maximumstable set in G. By Ω(G) we mean the family of all maximum stable sets ofthe graph G.

A set A ⊆ V (G) is a local maximum stable set of G if A is a maximumstable set in the subgraph spanned by N [A], i.e., A ∈ Ω(G[N [A]]), [6]. LetΨ(G) stand for the family of all local maximum stable sets of G. In [10],Nemhauser and Trotter Jr. showed that any local maximum stable set of agraph can be enlarged to one of its maximum stable sets.

A matching in a graph G = (V,E) is a set of edges M ⊆ E such that notwo edges of M share a common vertex. A perfect matching is a matchingsaturating all the vertices of the graph.

A cycle C is M-alternating if for any two incident edges of C exactly oneof them belongs to the matching M (see Kroghdal [5]). It is clear that anM -alternating cycle should be of even size.

A matching M in G is called alternating cycle-free if G has no M -alternatingcycle. Alternating cycle-free matchings for bipartite graphs were first definedby Kroghdal in [5]. This kind of matchings was also investigated in connectionwith the so-called jump-number problem for partially ordered sets (see Chatyand Chein [1], Muller [9], Lozin and Gerber [8]).

A matching M = {aibi : ai, bi ∈ V (G), 1 ≤ i ≤ k} of a graph G is called auniquely restricted matching if M is the unique perfect matching of the sub-graph of G induced by {ai, bi : 1 ≤ i ≤ k}, (see Golumbic, Hirst, Lewenstein[2]). It appears also in the context of matrix theory, as a constrained matching(see Hershkowitz and Schneider [3]).

The matching number of G, denoted by μ(G), is the cardinality of amaximum matching (i.e., of a matching of maximum size). By μr(G) wemean the size of a maximum uniquely restricted matching of G. Clearly,0 ≤ μr(G) ≤ μ(G) holds for any graph G.

Since any forest G, by definition, has no cycles, Theorem 1.1 ensures thatall matchings of a forest are uniquely restricted, and therefore, μr(G) = μ(G).

V.E. Levit, E. Mandrescu / Electronic Notes in Discrete Mathematics 22 (2005) 261–265262

Theorem 1.1 [2] A matching M in a graph G is uniquely restricted if andonly if G does not contain an alternating cycle with respect to M .

A greedoid (see Korte, Lovasz, Schrader [4]) is a pair (E,F), where F ⊆ 2E

is a set system satisfying the following conditions:

(Accessibility) for every non-empty X ∈ F , there is some x ∈ X such thatX − {x} ∈ F ;

(Exchange) for X,Y ∈ F , |X| = |Y | + 1, there is some x ∈ X − Y such thatY ∪ {x} ∈ F .

Theorem 1.2 [7] For a bipartite graph G, the family Ψ(G) is a greedoid ifand only if all maximum matchings of G are uniquely restricted.

In [2], Golumbic, Hirst and Lewenstein observed that μr(G) = μ(G) forany graph G having no cycle or with only odd cycles. They posed the problemof finding other graphs enjoying this equality.

In this paper we describe unicycle graphs with uniquely restricted max-imum matchings. Moreover, we characterize unicycle bipartite graphs satis-fying a stronger condition, namely, having all maximum matchings uniquelyrestricted.

2 Results

Lemma 2.1 If G is a bipartite graph having C as its unique cycle and M is amatching, then M is uniquely restricted if and only if |M ∩ E(C)| < |E(C)| /2.

If G is a non-unicycle bipartite graph, then the difference μ(G) − μr(G)can be arbitrarily large; for instance, μ(Kn,n) = n, while μr(Kn,n) = 1, whereKn,n designates the complete bipartite graph with 2n vertices.

Corollary 2.2 If G is a unicycle bipartite graph, then μ(G) − μr(G) ≤ 1.

The next result characterizes unicycle bipartite graphs having uniquelyrestricted maximum matchings.

Theorem 2.3 If G is a bipartite graph having C as its unique cycle, then thefollowing assertions are equivalent:

(i) μr(G) = μ(G);

(ii) there is a tree T of the forest G − C and some w ∈ V (T ), so thatμ(T − w) = μ(T ), where vw ∈ E(G) − E(C) for some v ∈ V (C);

(iii) there is an edge vw in a maximum matching with |{v, w} ∩ V (C)| = 1.

V.E. Levit, E. Mandrescu / Electronic Notes in Discrete Mathematics 22 (2005) 261–265 263

Theorem 2.3 helps us to build an algorithm recognizing unicycle bipartitegraphs owning uniquely restricted maximum matchings.

Let us notice that if a graph has a uniquely restricted maximum matching,it does not imply that all its maximum matchings are uniquely restricted. Inother words, the assertions (ii) and (iii) in Theorem 2.3 do not guarantee thatthe graph has only uniquely restricted maximum matchings.

Proposition 2.4 For any graph G and its edge e = xy, the following asser-tions are equivalent:

(i) e belongs to no maximum matching of G;

(ii) any maximum matching saturates x and y by two edges;

(iii) μ(G − U) = μ(G) − 2, where U contains all the edges of G incidentto x or y.

If C is the unique cycle of the bipartite graph G, then

μ(G − C) + |E(C)| /2 ≤ μ(G).

Let us remark that the equality μ(G − C) + |E(C)| /2 = μ(G) shows thatG has at least one non-uniquely restricted maximum matching.

In the following theorem we characterize unicycle bipartite graphs thathave only uniquely restricted maximum matchings.

Theorem 2.5 If G is a bipartite graph having C as its unique cycle, then thefollowing assertions are equivalent:

(i) there is e ∈ E(C) such that no maximum matching of G contains e;

(ii) all maximum matchings of G are uniquely restricted;

(iii) μ(G − C) + |E(C)| /2 < μ(G);

(iv) μ(G−U) = μ(G)− 2, where U contains all the edges of G incident tothe endpoints of some edge belonging to C;

(v) there is no maximum matching M such that C is a M-alternating cycle;

(vi) Ψ(G) is a greedoid;

(vii) each maximum matching has an edge e = xy with |{x, y} ∩ V (C)| = 1;

(viii) there are at least two vertices x, y on C such that the number of edgeson C separating x, y is odd and μ(Tv − wv) = μ(Tv) holds for any v ∈ {x, y},where Tv denotes a tree of G − C joined to the vertex v of C by the edge vwv

with wv ∈ V (Tv).

Theorem 2.5 helps us to build an algorithm for recognizing a unicyclebipartite graph G having only uniquely restricted maximum matchings, thathas time complexity O(|V (G)|).

V.E. Levit, E. Mandrescu / Electronic Notes in Discrete Mathematics 22 (2005) 261–265264

Theorem 1.1 shows that non-bipartite unicycle graphs own only uniquelyrestricted maximum matchings. Consequently, to check whether all maximummatchings of a unicycle graph are uniquely restricted, one has first to recognizea unicycle graph, then to exit if its cycle is of odd size, and to continue if itscycle is of even size, treating it as a unicycle bipartite graph.

References

[1] Chaty, G., and M. Chein, Ordered matchings and matchings without alternatingcycles in bipartite graphs, Utilitas Mathematica 16 (1979) 183-187.

[2] Golumbic, M. C., T. Hirst, and M. Lewenstein, Uniquely restricted matchings,Algorithmica 31 (2001) 139-154.

[3] Hershkowitz, D., and H. Schneider, Ranks of zero patterns and sign patterns,Linear and Multilinear Algebra 34 (1993) 3-19.

[4] Korte, B., L. Lovasz, and R. Schrader, “Greedoids,” Springer-Verlag, Berlin,1991.

[5] Krogdahl, S., The dependance graph for bases in matroids, DiscreteMathematics 19 (1977) 47-59.

[6] Levit, V. E., and E. Mandrescu, A new greedoid: the family of local maximumstable sets of a forest, Discrete Applied Mathematics 124 (2002) 91-101.

[7] Levit, V. E., and E. Mandrescu, Local maximum stable sets in bipartite graphswith uniquely restricted maximum matchings, Discrete Applied Mathematics132 (2003) 163-174.

[8] Lozin, V. V., and M. U. Gerber, On the jump number problem in hereditaryclasses of bipartite graphs, Order 17 (2000) 377-385.

[9] Muller, H., Alternating cycle-free matchings, Order 7 (1990) 11-21.

[10] Nemhauser, G. L., and L. E. Trotter, Jr., Vertex packings: structural propertiesand algorithms, Mathematical Programming 8 (1975) 232-248.

V.E. Levit, E. Mandrescu / Electronic Notes in Discrete Mathematics 22 (2005) 261–265 265