On Minimum Maximal Distance-k Matchings

Kartynnik, Yury; Ryzhikov, Andrew

doi:10.1016/j.endm.2016.11.011

Cited by 6 publications

(5 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A distance-based extension of equimatchability was recently studied by Kartynnik and Ryzhikov [15]. In this paper we generalize the property of equimatchability of graphs in two further ways by introducing two graph parameters measuring how far a graph is from being equimatchable.…”

Section: Introductionmentioning

confidence: 99%

On two extensions of equimatchable graphs

Deniz

Ekim

Hartinger

et al. 2017

Discrete Optimization

View full text Add to dashboard Cite

A graph is said to be equimatchable if all its maximal matchings are of the same size. In this work we introduce two extensions of the property of equimatchability by defining two new graph parameters that measure how far a graph is from being equimatchable. The first one, called the matching gap, measures the difference between the sizes of a maximum matching and a minimum maximal matching. The second extension is obtained by introducing the concept of equimatchable sets; a set of vertices in a graph G is said to be equimatchable if all maximal matchings of G saturating the set are of the same size. Noting that G is equimatchable if and only if the empty set is equimatchable, we study the equimatchability defect of the graph, defined as the minimum size of an equimatchable set in it. We develop several inapproximability and parameterized complexity results and algorithms regarding the computation of these two parameters, a characterization of graphs of unit matching gap, exact values of the equimatchability defect of cycles, and sharp bounds for both parameters.

show abstract

Section: Introductionmentioning

confidence: 99%

On two extensions of equimatchable graphs

Deniz

Ekim

Hartinger

et al. 2017

Discrete Optimization

View full text Add to dashboard Cite

show abstract

“…A graph G is called equimatchable if every maximal matching of G has the same cardinality. The structure of equimatchable graphs has been widely studied in the literature (see for instance [1,4,2,5,7,8,10,14,15,16]). The counterpart of equimatchable graphs for independent sets is called well-covered graphs: a graph is well-covered if all its maximal independent sets have the same size.…”

Section: Introductionmentioning

confidence: 99%

Critical Equimatchable Graphs

Deniz¹,

Ekim²

2022

Preprint

View full text Add to dashboard Cite

A graph G is equimatchable if every maximal matching of G has the same cardinality. In this paper, we investigate equimatchable graphs such that the removal of any edge harms the equimatchability, called edge-critical equimatchable graphs (ECE-graphs). We show that apart from two simple cases, namely bipartite ECE-graphs and even cliques, all ECE-graphs are 2-connected factor-critical. Accordingly, we give a characterization of factor-critical ECE-graphs with connectivity 2. Our result provides a partial answer to an open question posed by Levit and Mandrescu [12] on the characterization of well-covered graphs with no shedding vertex. We also introduce equimatchable graphs such that the removal of any vertex harms the equimatchability, called vertex-critical equimatchable graphs (VCE-graphs). To conclude, we enlighten the relationship between various subclasses of equimatchable graphs (including ECE-graphs and VCE-graphs) and discuss the properties of factor-critical ECE-graphs with connectivity at least 3.

show abstract

“…Moreover, a distance-2 matching is usually called an induced matching. These invariants have been studied in various papers and are closely related to many other concepts (see [1,2,6,7,8,9,11,12] for examples). The magnitude relation between the beans function B G (x) for a given value of x ≥ 1 and each of the two invariants will be explained in detail in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Recursive Formulas for Beans Functions of Graphs

Enami¹,

Negami²

2020

tag

View full text Add to dashboard Cite

In this paper, we regard each edge of a connected graph G as a line segment having a unit length, and focus on not only the "vertices" but also any "point" lying along such a line segment. So we can define the distance between two points on G as the length of a shortest curve joining them along G. The beans function B G (x) of a connected graph G is defined as the maximum number of points on G such that any pair of points have distance at least x > 0. We shall show a recursive formula for B G (x) which enables us to determine the value of B G (x) for all x ≤ 1 by evaluating it only for 1/2 < x ≤ 1. As applications of this recursive formula, we shall propose an algorithm for computing B G (x) for a given value of x ≤ 1, and determine the beans functions of the complete graphs K n .

show abstract

On Minimum Maximal Distance-k Matchings

Cited by 6 publications

References 19 publications

On two extensions of equimatchable graphs

On two extensions of equimatchable graphs

Critical Equimatchable Graphs

Recursive Formulas for Beans Functions of Graphs

Contact Info

Product

Resources

About