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On two equimatchable graph classes

Ken-ichi Kawarabayashi, Michael D. Plummer, Akira Saito
2003 Discrete Mathematics  
In this paper, we use the Gallai-Edmonds decomposition theory for matchings to determine the equimatchable members of two important graph classes.  ...  We ÿnd that there are precisely 23 3-connected planar graphs (i.e., 3-polytopes) which are equimatchable and that there are only two cubic equimatchable graphs.  ...  Acknowledgements The authors wish to thank one of the referees for spotting a aw in our original proof of Lemma 1.10.  ... 
doi:10.1016/s0012-365x(02)00813-0 fatcat:flkrk7vp2zgt3ea2fyzf4hwh4a

Edge-Stable Equimatchable Graphs [article]

Zakir Deniz, Tınaz Ekim
2018 arXiv   pre-print
We are interested in equimatchable graphs such that the removal of any edge from the graph preserves the equimatchability.  ...  We call an equimatchable graph G edge-stable if G∖e, that is the graph obtained by the removal of edge e from G, is also equimatchable for any e ∈ E(G).  ...  We note that equimatchable graphs are closely related to a well-studied graph class called well-covered graphs.  ... 
arXiv:1602.09127v3 fatcat:ustumxxivzbojdckroqjfgygli

Critical Equimatchable Graphs [article]

Zakir Deniz, Tınaz Ekim
2022 arXiv   pre-print
We show that apart from two simple cases, namely bipartite ECE-graphs and even cliques, all ECE-graphs are 2-connected factor-critical.  ...  Our result provides a partial answer to an open question posed by Levit and Mandrescu on the characterization of well-covered graphs with no shedding vertex.  ...  For each class, one can check that the removal of no independent set of size 3 leaves a graph with a perfect matching, proving equimatchability by Lemma 2.6.  ... 
arXiv:2202.07929v1 fatcat:vypbbeuc3fdl7hgsk7pwuhrvba

Triangle-Free Equimatchable Graphs [article]

Yasemin Büyükçolak, Didem Gözüpek, Sibel Özkan
2021 arXiv   pre-print
A graph is called equimatchable if all of its maximal matchings have the same size. Frendrup et al. [8] provided a characterization of equimatchable graphs with girth at least 5.  ...  In this paper, we extend this result by providing a complete structural characterization of equimatchable graphs with girth at least 4, i.e., equimatchable graphs with no triangle, by identifying the equimatchable  ...  We then discuss the class of factor-critical equimatchable triangle-free graphs. In Section 4, we structurally characterize this graph class by identifying graph families in this class.  ... 
arXiv:1807.09520v2 fatcat:kd4ewissybe3fhgkgy7btxa23a

Linear Time Recognition of Equimatchable Split Graphs [article]

Mehmet Akif Yıldız
2019 arXiv   pre-print
On the other hand, G is called as a split graph if its vertices can be partitioned into two subsets for which one of them forms a clique whereas the second forms an independent set.  ...  We will give a linear time algorithm for recognition of equimatchable split graphs.  ...  Prof. at Bogaziçi University) for introducing me equimatchable graphs and giving me this problem.  ... 
arXiv:1911.04277v1 fatcat:qjyyowg5zffzzb657q7b7wbwnm

Equimatchable factor-critical graphs and independence number 2 [article]

Eduard Eiben, Michal Kotrbcik
2015 arXiv   pre-print
It is known that any 2-connected equimatchable graph is either bipartite, or factor-critical, and that these two classes are disjoint.  ...  On the other hand, since every 2-connected odd graph with independence number 2 is equimatchable, we get the following result.  ...  Equimatchable graphs G such that G ∪ {e} is equimatchable for every edge e of the complement G of G are further investigated in [3] , together with other extremal classes of equimatchable graphs.  ... 
arXiv:1501.07549v1 fatcat:5a3m3yoh6zbsxlifaxc5d5b2km

On two extensions of equimatchable graphs [article]

Zakir Deniz, Tınaz Ekim, Tatiana Romina Hartinger, Martin Milanič, Mordechai Shalom
2016 arXiv   pre-print
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.  ...  A graph is said to be equimatchable if all its maximal matchings are of the same size.  ...  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.  ... 
arXiv:1612.08531v1 fatcat:t5czsuzyk5e3zgzfddi3ksnnbu

Equimatchable bipartite graphs

Yasemin Büyükcolak, Didem Gözüpek, Sibel Özkan
2020 Discussiones Mathematicae Graph Theory  
[A note on equimatchable graphs, Australas. J.  ...  For an equimatchable graph, an edge is said to be a critical-edge if the graph obtained by the removal of this edge is not equimatchable.  ...  The class of equimatchable graphs is separated into two complementary subclasses, namely factor-critical and non-factor-critical equimatchable graphs.  ... 
doi:10.7151/dmgt.2356 fatcat:msuou63swzfvdkqqdbopditip4

Totally equimatchable graphs

Jerzy Topp, Preben D. Vestergaard
1997 Discrete Mathematics  
A subset X of vertices and edges of a graph G is totally matching if no two elements of X are adjacent or incident.  ...  In this paper we determine all graphs in which every maximal total matching is maximum.  ...  A connected graph is totally equimatchable if and only if it is one of the graphs Claim 1. Let M be a maximum matching of G.  ... 
doi:10.1016/s0012-365x(96)00062-3 fatcat:cqia5za3mvctbdg6let2hhh2ci

Equimatchable claw-free graphs

Saieed Akbari, Hadi Alizadeh, Tınaz Ekim, Didem Gözüpek, Mordechai Shalom
2018 Discrete Mathematics  
In this paper, we provide, to the best of our knowledge, the first characterization of claw-free equimatchable graphs by identifying the equimatchable claw-free graph families.  ...  A graph is equimatchable if all of its maximal matchings have the same size. A graph is claw-free if it does not have a claw as an induced subgraph.  ...  subgraph of them; otherwise, H has to be compared to each one of these graphs and their relevant subgraphs, where each comparison takes constant time.  ... 
doi:10.1016/j.disc.2018.06.043 fatcat:krsy7k6bxfgwrkmq7hqmyzezre

2-connected equimatchable graphs on surfaces [article]

Eduard Eiben, Michal Kotrbčík
2013 arXiv   pre-print
We use this result to improve the upper bounds on the maximum size of $2$-connected equimatchable factor-critical graphs embeddable in the orientable surface of genus $g$ to $4\sqrt g+17$ if $g\le 2$ and  ...  A graph $G$ is equimatchable if any matching in $G$ is a subset of a maximum-size matching. It is known that any $2$-connected equimatchable graph is either bipartite or factor-critical.  ...  Therefore, these two classes are disjoint. All 3-connected planar equimatchable graphs are characterised in [7] -there are 23 such graphs and none of them is bipartite.  ... 
arXiv:1312.3423v1 fatcat:747xzsp5f5hp7bdhi5cddgazei

Page 6786 of Mathematical Reviews Vol. , Issue 87m [page]

1987 Mathematical Reviews  
The author further shows that if G is an equimatchable factor-critical graph having connectivity 2 and vertex cut-set S = {s,,82}, then G—S has exactly two components, one of even order and the other of  ...  Isolated vertices or loops are circuits with one vertex, and edges or two vertices joined by a pair of edges are circuits with two vertices.  ... 

On Minimum Maximal Distance-k Matchings

Yury Kartynnik, Andrew Ryzhikov
2016 Electronic Notes in Discrete Mathematics  
We introduce the class of k-equimatchable graphs which is an edge analogue of k-equipackable graphs. We prove that the recognition of k-equimatchable graphs is co-NP-complete for any fixed k ≥ 2.  ...  We provide a simple characterization for the class of strongly chordal graphs with equal k-packing and k-domination numbers.  ...  We also thank anonymous reviewers for their numerous helpful comments on the preliminary versions of this paper.  ... 
doi:10.1016/j.endm.2016.11.011 fatcat:j3gvr6p2anbtdalx6sid6lqgki

On Minimum Maximal Distance-k Matchings [article]

Yury Kartynnik, Andrew Ryzhikov
2017 arXiv   pre-print
We introduce the class of k-equimatchable graphs which is an edge analogue of k-equipackable graphs. We prove that the recognition of k-equimatchable graphs is co-NP-complete for any fixed k > 2.  ...  We provide a simple characterization for the class of strongly chordal graphs with equal k-packing and k-domination numbers.  ...  We also thank anonymous reviewers for their numerous helpful comments on the preliminary versions of this paper.  ... 
arXiv:1602.04581v4 fatcat:oxeev4lkzvcwze4hwht5clk4ya

Well-indumatched Trees and Graphs of Bounded Girth [article]

S. Akbari, T. Ekim, A.H. Ghodrati, S. Zare
2019 arXiv   pre-print
We also show that, although the recognition of well-indumatched graphs is known to be co-NP-complete in general, one can recognize in polynomial time well-indumatched graphs where the size of maximal induced  ...  On the other hand, there are infinitely many well-indumatched unicyclic graphs of girth k, where k is in 3, 5, 7 or k is an even integer greater than 2.  ...  Some other graph classes that could be investigated in this direction are interval graphs, claw-free graphs or equimatchable graphs.  ... 
arXiv:1903.03197v2 fatcat:uqxica22f5fmzminvnjjk7za5y
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