On the Maximum Uniquely Restricted Matching for Bipartite Graphs

Discrete Applied Mathematics, 2011

We present a collection of new structural, algorithmic, and complexity results for two types of matching problems. The first problem involves the computation of k-maximal matchings, where a matching is k-maximal if it admits no augmenting path with ≤ 2k vertices. The second involves finding a maximal set of vertices that is matchable-comprising one side of the edges in some matching. Among our results, we prove that the minimum cardinality β 2 of a 2-maximal matching is at most the minimum cardinality µ of a maximal matchable set, with equality attained for triangle-free graphs. We show that the parameters β 2 and µ are NP-hard to compute in bipartite and chordal graphs, but can be computed in linear time on a tree. Finally, we also give a simple linear-time algorithm for finding a 3-maximal matching, a consequence of which is a simple linear-time 3/4-approximation algorithm for the maximum-cardinality matching problem in a general graph.

Discussiones Mathematicae Graph Theory, 2013

A semi-matching in a bipartite graph G = (U ∪ V, E) is a set of edges 11 M ⊆ E such that each vertex in U is incident with exactly one edge in M . A 12 set M ⊆ E(G) is an (f, g)-quasi-matching of G = (U ∪ V, E) if every element 13 u of U has at most f (u) incident edges from M and every element v of V 14 has at least g(v) incident edges from M . In this paper we give an algorithm 15 that for a graph with n ≤ m constructs an (f, g)-quasi-matching in running 16 time O(m · v∈V g(v)). This result supports the result from [5] related 17 to a computing an optimal semi-matching in running time O( √ nm log n).

Scientific Annals of Computer Science

The problem of computing a maximum weight matching in a bipartite graph is one of the fundamental algorithmic problems that has played an important role in the development of combinatorial optimization and algorithmics. Let G w,σ is a collection of all weighted bipartite graphs, each having σ and w as the size of each of the non-empty subset of the vertex partition and the total weight of the graph, respectively. We give a tight lower bound w-σ σ + 1 for the set {Wt(mwm(G)) | G ∈ G w,σ } which denotes the collection of weights of maximum weight bipartite matchings of all the graphs in G w,σ .

Discrete Applied Mathematics, 2011

We present a collection of new structural, algorithmic, and complexity results for two types of matching problems. The first problem involves the computation of k-maximal matchings, where a matching is k-maximal if it admits no augmenting path with ≤ 2k vertices. The second involves finding a maximal set of vertices that is matchable -comprising one side of the edges in some matching. Among our results, we prove that the minimum cardinality β 2 of a 2-maximal matching is at most the minimum cardinality µ of a maximal matchable set, with equality attained for triangle-free graphs. We show that the parameters β 2 and µ are NP-hard to compute in bipartite and chordal graphs, but can be computed in linear time on a tree. Finally, we also give a simple linear-time algorithm for finding a 3-maximal matching, a consequence of which is a simple linear-time 3/4-approximation algorithm for the maximum-cardinality matching problem in a general graph.

Discrete Applied Mathematics, 2003

Given a simple bipartite graph G and an integer t ≥ 2, we derive a formula for the maximum number of edges in a subgraph H of G so that H contains no node of degree larger than t and H contains no complete bipartite graph K t,t as a subgraph. In the special case t = 2 this fomula was proved earlier by Z. Király [6], sharpening a result of D. Hartvigsen [4]. For any integer t ≥ 2, we also determine the maximum number of edges in a subgraph of G that contains no complete bipartite graph, as a subgraph, with more than t nodes. The proofs are based on a general min-max result of [2] concerning crossing bi-supermodular functions.

SIAM Journal on Discrete Mathematics, 2017

Let f (n) be the smallest number such that every collection of n matchings, each of size at least f (n), in a bipartite graph, has a full rainbow matching. Generalizing famous conjectures of Ryser, Brualdi and Stein, Aharoni and Berger [1] conjectured that f (n) = n + 1 for every n > 1. Clemens and Ehrenmüller [4] proved that f (n) ≤ 3 2 n + o(n). We show that the o(n) term can be reduced to a constant, namely f (n) ≤ 3 2 n + 1.

Discrete Mathematics, 2012

For a graph G let L(G) and l(G) denote the size of the largest and smallest maximum matching of a graph obtained from G by removing a maximum matching of G. We show that L(G) ≤ 2l(G), and L(G) ≤ 3 2 l(G) provided that G contains a perfect matching. We also characterize the class of graphs for which L(G) = 2l(G). Our characterization implies the existence of a polynomial algorithm for testing the property L(G) = 2l(G). Finally we show that it is NP-complete to test whether a graph G containing a perfect matching satisfies L(G) = 3 2 l(G).

Given an undirected bipartite graph G = (A ∪ B, E), the b-matching of G matches each vertex v in A (B) to at least 1 and at most b(v) vertices in B (A), where b(v) denotes the capacity of v. In this paper, we present an O(n 3) time algorithm for finding a maximum weight b-matching of G, where |A| + |B| = O(n). Our algorithm improves the previous best time complexity of O(n 3 log n) for this problem.

Annals of Operations Research, 2021

Given a graph G = (V , E), an edge bipartization set of G is a subset E ⊆ E(G) such that G = (V , E\E) is bipartite. An edge bipartization set that is also a matching of G is called a bipartizing matching of G. Let BM be the family of all graphs admitting a bipartizing matching. Although every graph has an edge bipartization set, the problem of recognizing graphs G having bipartizing matchings (G ∈ BM) is challenging. A (k, d)-coloring of a graph G is a k-coloring of V (G) such that each vertex of G has at most d neighbors with the same color as itself. Clearly a (k, 0)-coloring is a proper vertex k-coloring of G and, for any d > 0, the k-coloring is non-proper, also known as defective. A graph belongs to BM if and only if it admits a (2, 1)-coloring. Lovász (1966) proved that for any integer k > 0, any graph of maximum degree Δ admits a (k, Δ/k)-coloring. In this paper, we show that it is NP-complete to determine whether a 3-colorable planar graph of maximum degree 4 belongs to BM , i.e., (2, 1)-colorable. Besides, we show that it is NP-complete to determine whether planar graphs of maximum degree 6 or 8 admit a (2, 2) or (2, 3)-coloring, respectively. Due to Lovász (1966), our results are tight in the sense that on graphs with maximum degree three, five, and seven, there always exists a (2, 1), (2, 2), and (2, 3)-coloring, respectively. Finally, we present polynomial-time algorithms for particular graph classes, besides some remarks on the parameterized complexity of the problem of recognizing graphs in BM .

International Symposium on Algorithms and Computation, 2001

In this paper, we study bounds on maximal and maximum matchings in special graph classes, specifically triangulated graphs and graphs with bounded maximum degree. For each class, we give a lower bound on the size of matchings, and prove that it is tight for some graph within the class.