Small maximal matchings of random cubic graphs

We consider the expected size of a smallest maximal matching of cubic graphs. Firstly, we present a randomized greedy algorithm for finding a small maximal matching of cubic graphs. We analyze the average‐case performance of this heuristic on random n‐vertex cubic graphs using differential equations. In this way, we prove that the expected size of the maximal matching returned by the algorithm is asymptotically almost surely (a.a.s.) less than 0.34623n. We also give an existence proof which shows that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. less than 0.3214n. It is known that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. larger than 0.3158n. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 293–323, 2009     

On randomized greedy matchings

We analyze a randomized greedy matching algorithm (RGA) aimed at producing a matching with a large number of edges in a given weighted graph. RGA was first introduced and studied by Dyer and Frieze in [3] for unweighted graphs. In the weighted version, at each step a new edge is chosen from the remaining graph with probability proportional to its weight, and is added to the matching. The two vertices of the chosen edge are removed, and the step is repeated until there are no edges in the remaining graph. We analyze the expected size μ(G) of the number of edges in the output matching produced by RGA, when RGA is repeatedly applied to the same graph G. Let r(G)=μ(G)/m(G), where m(G) is the maximum number of edges in a matching in G. For a class 𝒢 of graphs, let ρ(𝒢) be the infimum values r(G) over all graphs G in 𝒢 (i.e., ρ is the “worst” performance ratio of RGA restricted to the class 𝒢). Our main results are bound for μ, r, and ρ. For example, the following results improve or generalize similar results obtained in [3] for the unweighted version of RGA; \begin{eqnarray*}r(G)&\ge&{1\over 2-|V|/2|E|}\quad \mbox{(if $G$ has a perfect matching)}\\ {\sqrt{26}-4\over 2}&\le&\rho(\hbox{\sf SIMPLE PLANAR GRAPHS})\le.68436349\\ \rho(\hbox{SIMPLE $\Delta$-GRAPHS})&\ge&{1\over2}+{\sqrt{(\Delta-1)^2+1}-(\Delta-1)\over2}\end{eqnarray*} (where the class is the set of graphs of maximum degree at most Δ). © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10 : 353–383, 1997     

An Even Faster and More Unifying Algorithm for Comparing Trees via Unbalanced Bipartite Matchings

A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure has only been concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with all their nodes labeled with the same symbol) and evolutionary trees (i.e., leaf-labeled trees with distinct symbols for distinct leaves). This paper presents an algorithm for comparing trees that are labeled in an arbitrary manner. In addition to this generality, this algorithm is faster than the previous algorithms.Another contribution of this paper is on maximum weight bipartite matchings. We show how to speed up the best known matching algorithms when the input graphs are node-unbalanced or weight-unbalanced. Based on these enhancements, we obtain an efficient algorithm for a new matching problem called the hierarchical bipartite matching problem, which is at the core of our maximum agreement subtree algorithm.     

A Planar linear arboricity conjecture

The linear arboricity la(G) of a graph G is the minimum number of linear forests (graphs where every connected component is a path) that partition the edges of G. In 1984, Akiyama et al. [Math Slovaca 30 (1980), 405–417] stated the Linear Arboricity Conjecture (LAC) that the linear arboricity of any simple graph of maximum degree Δ is either ?Δ/2? or ?(Δ + 1)/2?. In [J. L. Wu, J Graph Theory 31 (1999), 129–134; J. L. Wu and Y. W. Wu, J Graph Theory 58(3) (2008), 210–220], it was proven that LAC holds for all planar graphs. LAC implies that for Δ odd, la(G) = ?Δ/2?. We conjecture that for planar graphs, this equality is true also for any even Δ?6. In this article we show that it is true for any even Δ?10, leaving open only the cases Δ = 6, 8. We present also an O(n logn) algorithm for partitioning a planar graph into max{la(G), 5} linear forests, which is optimal when Δ?9. © 2010 Wiley Periodicals, Inc. J Graph Theory     

Connected matchings in chordal bipartite graphs

A connected matching in a graph is a collection of edges that are pairwise disjoint but joined by another edge of the graph. Motivated by applications to Hadwiger’s conjecture, Plummer, Stiebitz, and Toft (2003) introduced connected matchings and proved that, given a positive integer $k$, determining whether a graph has a connected matching of size at least $k$ is NP-complete. Cameron (2003) proved that this problem remains NP-complete on bipartite graphs, but can be solved in polynomial-time on chordal graphs. We present a polynomial-time algorithm that finds a maximum connected matching in a chordal bipartite graph. This includes a novel edge-without-vertex-elimination ordering of independent interest. We give several applications of the algorithm, including computing the Hadwiger number of a chordal bipartite graph, solving the unit-time bipartite margin-shop scheduling problem in the case in which the bipartite complement of the precedence graph is chordal bipartite, and determining–in a totally balanced binary matrix–the largest size of a square sub-matrix that is permutation equivalent to a matrix with all zero entries above the main diagonal.     

Optimum matching forests II: General weights

Matching forests generalize branchings in a directed graph and matchings in an undirected graph. We present an efficient algorithm, the PMF Algorithm, for the problem: given a mixed graphG and a real weight on each of its edges, find a perfect matching forest of maximum weight-sum. The PMF Algorithm proves the sufficiency of a linear system which definesP ⁼ (G) andP(G), the convex hull of incidence vectors of perfect matching forests and matching forests respectively ofG. The algorithm also provides a generalization of Tutte's theorem on the existence of perfect matchings in an undirected graph.Research partially supported by a N.R.C. of Canada Postdoctorate Fellowship.     

Efficient Algorithms for Variants of Weighted Matching and Assignment Problems

Obtaining a matching in a graph satisfying a certain objective is an important class of graph problems. Matching algorithms have received attention for several decades. However, while there are efficient algorithms to obtain a maximum weight matching, not much is known about the maximum weight maximum cardinality, and maximum cardinality maximum weight matching problems for general graphs. Our contribution in this work is to show that for bounded weight input graphs one can obtain an algorithm for both maximum weight maximum cardinality (for real weights), and maximum cardinality maximum weight matching (for integer weights) by modifying the input and running the existing maximum weight matching algorithm. Also, given the current state of the art in maximum weight matching algorithms, we show that, for bounded weight input graphs, both maximum weight maximum cardinality, and maximum cardinality maximum weight matching have algorithms of similar complexities to that of maximum weight matching. Subsequently, we also obtain approximation algorithms for maximum weight maximum cardinality, and maximum cardinality maximum weight matching.      

Bounding the Size of Equimatchable Graphs of Fixed Genus

A graph G is said to be equimatchable if every matching in G extends to (i.e., is a subset of) a maximum matching in G. In an earlier paper with Saito, the authors showed that there are only a finite number of 3-connected equimatchable planar graphs. In the present paper, this result is extended by showing that in a surface of any fixed genus (orientable or non-orientable), there are only a finite number of 3-connected equimatchable graphs having a minimal embedding of representativity at least three. (In fact, if the graphs considered are non-bipartite, the representativity three hypothesis may be dropped.) The proof makes use of the Gallai-Edmonds decomposition theorem for matchings.      

Unicycle graphs and uniquely restricted maximum matchings

A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. G is a unicycle graph if it owns only one cycle. Golumbic, Hirst and Lewenstein observed that for a tree or a graph with only odd cycles the size of a maximum uniquely restricted matching is equal to the matching number of the graph. In this paper we characterize unicycle graphs enjoying this equality. Moreover, we describe unicycle graphs with only uniquely restricted maximum matchings. Using these findings, we show that unicycle graphs having only uniquely restricted maximum matchings can be recognized in polynomial time.     

Matching graphs

The matching graph M(G) of a graph G is that graph whose vertices are the maximum matchings in G and where two vertices M₁ and M₂ of M(G) are adjacent if and only if |M₁ − M₂| = 1. When M(G) is connected, this graph models a metric space whose metric is defined on the set of maximum matchings in G. Which graphs are matching graphs of some graph is not known in general. We determine several forbidden induced subgraphs of matching graphs and add even cycles to the list of known matching graphs. In another direction, we study the behavior of sequences of iterated matching graphs. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 73–86, 1998     

Minimum Color Sum of Bipartite Graphs

The problem ofminimum color sumof a graph is to color the vertices of the graph such that the sum (average) of all assigned colors is minimum. Recently it was shown that in general graphs this problem cannot be approximated withinn^{1 − ε}, for any ε > 0, unlessNP = ZPP(Bar-Noyet al., Information and Computation140(1998), 183–202). In the same paper, a 9/8-approximation algorithm was presented for bipartite graphs. The hardness question for this problem on bipartite graphs was left open. In this paper we show that the minimum color sum problem for bipartite graphs admits no polynomial approximation scheme, unlessP = NP. The proof is byL-reducing the problem of finding the maximum independent set in a graph whose maximum degree is four to this problem. This result indicates clearly that the minimum color sum problem is much harder than the traditional coloring problem, which is trivially solvable in bipartite graphs. As for the approximation ratio, we make a further step toward finding the precise threshold. We present a polynomial 10/9-approximation algorithm. Our algorithm uses a flow procedure in addition to the maximum independent set procedure used in previous solutions.     

A fast algorithm for coloring Meyniel graphs

We color the nodes of a graph by first applying successive contractions to non-adjacent nodes until we get a clique; then we color the clique and decontract the graph. We show that this algorithm provides a minimum coloring and a maximum clique for any Meyniel graph by using a simple rule for choosing which nodes are to be contracted. This O(n³) algorithm is much simpler than those already existing for Meyniel graphs. Moreover, the optimality of this algorithm for Meyniel graphs provides an alternate nice proof of the following result of Hoàng: a graph G is Meyniel if and only if, for any induced subgraph of G, each node belongs to a stable set that meets all maximal cliques. Finally we give a new characterization for Meyniel graphs.     

A characterization of chain probe graphs

A chain probe graph is a graph that admits an independent set S of vertices and a set F of pairs of elements of S such that G+F is a chain graph (i.e., a 2K ₂-free bipartite graph). We show that chain probe graphs are exactly the bipartite graphs that do not contain as an induced subgraph a member of a family of six forbidden subgraphs, and deduce an O(n ²) recognition algorithm.     

Total Dual Integrality of Matching Forest Constraints

Let G=(V, E, A) be a mixed graph. That is, (V, E) is an undirected graph and (V, A) is a directed graph. A matching forest (introduced by R. Giles) is a subset F of EèAE\cup A such that F contains no circuit (in the underlying undirected graph) and such that for each v ? Vv\in V there is at most one e ? Fe\in F such that v is head of e. (For an undirected edge e, both ends of e are called head of e.) Giles gave a polynomial-time algorithm to find a maximum-weight matching forest, yielding as a by-product a characterization of the inequalities determining the convex hull of the incidence vectors of the matching forests. We prove that these inequalities form a totally dual integral system. It is equivalent to an ``all-integer' min-max relation for the maximum weight of a matching forest. Our proof is based on an exchange property for matching forests, and implies Giles' characterization.     

Mixing of the Glauber dynamics for the ferromagnetic Potts model

We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree (Δ) of the underlying graph and the number of colours or spins (q) in determining whether the dynamics mixes rapidly or not. We find a lower bound L on the number of colours such that Glauber dynamics is rapidly mixing if at least L colours are used. We give a closely‐matching upper bound U on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random Δ‐regular graphs when at most U colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree Δ, e.g. toroidal grids for Δ = 4. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 21–52, 2016     

Recognizing near-bipartite Pfaffian graphs in polynomial time

A matching covered graph is a non-trivial connected graph in which every edge is in some perfect matching. A non-bipartite matching covered graph G is near-bipartite if there are two edges e₁ and e₂ such that G−e₁−e₂ is bipartite and matching covered. In 2000, Fischer and Little characterized Pfaffian near-bipartite graphs in terms of forbidden subgraphs [I. Fischer, C.H.C. Little, A characterization of Pfaffian near bipartite graphs, J. Combin. Theory Ser. B 82 (2001) 175-222.]. However, their characterization does not imply a polynomial time algorithm to recognize near-bipartite Pfaffian graphs. In this article, we give such an algorithm.We define a more general class of matching covered graphs, which we call weakly near-bipartite graphs. This class includes the near-bipartite graphs. We give a polynomial algorithm for recognizing weakly near-bipartite Pfaffian graphs. We also show that Fischer and Little’s characterization of near-bipartite Pfaffian graphs extends to this wider class.     

A PTAS for minimum <Emphasis Type="Italic">d</Emphasis>-hop connected dominating set in growth-bounded graphs

In this paper, we design the first polynomial time approximation scheme for d-hop connected dominating set (d-CDS) problem in growth-bounded graphs, which is a general type of graphs including unit disk graph, unit ball graph, etc. Such graphs can represent majority types of existing wireless networks. Our algorithm does not need geometric representation (e.g., specifying the positions of each node in the plane) beforehand. The main strategy is clustering partition. We select the d-CDS for each subset separately, union them together, and then connect the induced graph of this set. We also provide detailed performance and complexity analysis.     

A generalization of simplicial elimination orderings

We consider the class of graphs where every induced subgraph possesses a vertex whose neighborhood has no P₄ and no 2K₂. We prove that Berge's Strong Perfect Graph Conjecture holds for such graphs. The class generalizes several well-known families of perfect graphs, such as triangulated graphs and bipartite graphs. Testing membership in this class and computing the maximum clique size for a graph in this class is not hard, but finding an optimal coloring is NP-hard. We give a polynomial-time algorithm for optimally coloring the vertices of such a graph when it is perfect. © 1996 John Wiley & Sons, Inc.     

On finding augmenting graphs

Method of augmenting graphs is a general approach to solve the maximum independent set problem. As the problem is generally NP-hard, no polynomial time algorithms are available to implement the method. However, when restricted to particular classes of graphs, the approach may lead to efficient solutions. A famous example of this type is the maximum matching algorithm: it finds a maximum matching in a graph G, which is equivalent to finding a maximum independent set in the line graph of G. In the particular case of line graphs, the method reduces to finding augmenting (alternating) chains. Recent investigations of more general classes of graphs revealed many more types of augmenting graphs. In the present paper we study the problem of finding augmenting graphs different from chains. To simplify this problem, we introduce the notion of a redundant set. This allows us to reduce the problem to finding some basic augmenting graphs. As a result, we obtain a polynomial time solution to the maximum independent set problem in a class of graphs which extends several previously studied classes including the line graphs.