Bicritical Graphs Research Articles

Minimally k-Factor-Critical Graphs for Some Large k

A graph G of order n is said to be k-factor-critical for integers $$1\le k < n$$ , if the removal of any k vertices results in a graph with a perfect matching. 1- and 2-factor-critical graphs are the well-known factor-critical and bicritical graphs, respectively. A k-factor-critical graph G is called minimal if for any edge $$e\in E(G)$$ , $$G-e$$ is not k-factor-critical. In 1998, O. Favaron and M. Shi conjectured that every minimally k-factor-critical graph of order n has the minimum degree $$k+1$$ and confirmed it for $$k=1, n-2, n-4$$ and $$n-6$$ . In this paper, we use a simple method to reprove the above results. As a main result, the further use of this method enables us to prove the conjecture to be true for $$k=n-8$$ . We also obtain that every minimally $$(n-6)$$ -factor-critical graph of order n has at most $$n-\Delta (G)$$ vertices with the maximum degree $$\Delta (G)$$ for $$\Delta (G)\ge n-4$$ .

Bicritical graphs without removable edges

A graph G with four or more vertices is bicritical if for any two distinct vertices x and y in G , G − x − y has a perfect matching. A graph is a brick if and only if it is 3-connected and bicritical. An edge e of a matching covered graph G is removable if G − e is also matching covered. Lovász (1987) showed that the only bricks without removable edges are K 4 and C 6 ¯ . In this paper, we present a characterization of bicritical graphs without removable edges by using the bricks and a type of 2-separation decomposition. Some properties of bicritical graphs without removable edges are revealed by using this characterization. Moreover, using this characterization, the claw-free bicritical graphs without removable edges are completely determined.

A Characterization of Domination Weak Bicritical Graphs with Large Diameter

The domination number of a graph G, denoted by $$\gamma (G)$$ , is the minimum cardinality of a dominating set of G. A vertex of a graph is called critical if its deletion decreases the domination number, and a graph is called critical if its all vertices are critical. A graph G is called weak bicritical if for every non-critical vertex $$x\in V(G)$$ , $$G-x$$ is a critical graph with $$\gamma (G-x)=\gamma (G)$$ . Furuya (Australas. J. Comb. 62:184–196, 2015) proved that the diameter of a connected weak bicritical graph G is at most $$2\gamma (G)-2$$ . In this paper, we characterize the connected weak bicritical graphs G whose diameter is exactly $$2\gamma (G)-2$$ . This is a generalization of some known results concerning the diameter of graphs with a domination-criticality.

4-Factor-Criticality of Vertex-Transitive Graphs

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-factor-critical graphs and 2-factor-critical graphs are well-known factor-critical graphs and bicritical graphs, respectively. It is known that if a connected vertex-transitive graph has odd order, then it is factor-critical, otherwise it is elementary bipartite or bicritical. In this paper, we show that a connected vertex-transitive non-bipartite graph of even order at least 6 is 4-factor-critical if and only if its degree is at least 5. This result implies that each connected non-bipartite Cayley graph of even order and degree at least 5 is 2-extendable.

Bicritical domination and double coalescence of graphs

Abstract A graph is bicritical if the removal of any pair of vertices decreases the domination number. We study the properties of bicritical graphs and their relation with critical graphs, and we obtain results for bicritical graphs with edge connectivity two or three. We also generalize the notion of the coalescence of two graphs and investigate the bicriticality of such graphs.

3‐Factor‐Criticality of Vertex‐Transitive Graphs

AbstractA graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle.

Small degree vertices in minimal bricks

Bricks are 3-connected bicritical graphs. Together with braces, they arise in matching theory as building blocks for matching covered graphs.We prove that every minimal brick on n vertices has at least n9 vertices of degree at most 4. This proves a relaxed version of a conjecture of Norine and Thomas, which suggests that every minimal brick has some fixed positive fraction of vertices of degree 3.

Constructing connected bicritical graphs with edge-connectivity 2

A graph G is said to be bicritical if the removal of any pair of vertices decreases the domination number of G. For a bicritical graph G with the domination number t, we say that G is t-bicritical. Let λ(G) denote the edge-connectivity of G. In [2], Brigham et al. (2005) posed the following question: If G is a connected bicritical graph, is it true that λ(G)≥3?In this paper, we give a negative answer toward this question; namely, we give a construction of infinitely many connected t-bicritical graphs with edge-connectivity 2 for every integer t≥5. Furthermore, we give some sufficient conditions for a connected 5-bicritical graph to have λ(G)≥3.

Factor-Critical Graphs with Given Number of Maximum Matchings

A connected graph G is said to be factor-critical if G − ν has a perfect matching for every vertex ν of G. In this paper, the factor-critical graphs G with |V(G)| maximum matchings and with |V(G)| + 1 ones are characterized, respectively. From this, some special bicritical graphs are characterized.

Construction for bicritical graphs and k-extendable bipartite graphs

A graph G is said to be bicritical if G - u - v has a perfect matching for every choice of a pair of points u and v . Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least 2 k + 2 points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an O ( mn ) algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G.

Bicritical domination

A graph G is domination bicritical if the removal of any pair of vertices decreases the domination number. Properties of bicritical graphs are studied. We show that a connected bicritical graph has domination number at least 3, minimum degree at least 3, and edge-connectivity at least 2. Ways of constructing a bicritical graph from smaller bicritical graphs are presented.

The enumeration of near-perfect matchings of factor-critical graphs

A matching of a graph is a near-perfect matching if it covers all but one vertex. A connected graph G is said to be factor-critical if G− v has perfect matchings for every vertex v of G. In this paper, the enumeration problems for near-perfect matchings in special factor-critical graphs are solved. From this, the exact values of the number of perfect matchings in special bicritical graphs are obtained.

Extendability and factor-criticality

A graph G of order n is p-factor-critical, where p is an integer of the same parity as n, if the removal of any set of p vertices of G results in a graph with a perfect matching. When n is even, G is p-extendable if any set of p independent edges is contained in a perfect matching. We show that for p even, every non-bipartite p-extendable graph is p-factor-critical, and every non-bipartite ( p+1)-extendable graph G is such that G− e is p-factor-critical for every edge e of G. This generalizes a result of Plummer on bicritical graphs. As a consequence, we get an upper bound on the independence number of a non-bipartite p-extendable graph.

From logical gates synthesis to chromatic bicritical clutters

The synthesis of logical gates by means of basic gates leads very naturally to the concept of clones of boolean functions and the corresponding lattice of clones largely studied by E. Post; a deep investigation into the clones of “positive functions” leads to the problem of characterizing their minimal functions; the canonical bijection between positive functions and Sperner hypergraphs (also called clutters) allows one to define (associated with these minimal functions) chromatic bicritical clutters which are nothing else than clutters H having the property that, for every edge e, the clutter whose edges are those of H disjoint from e, has a chromatic number two less than that of H. The bicritical clutters which are graphs are the source of a conjecture from Lovász (1966) which is still open: complete graphs are the only bicritical graphs! This survey is an attempt to grasp the (combinatorial and gate-theoretical) properties of these clutters, many of which inherit from some well-known properties of self-transversal clutters (which are exactly the non-trivial bicritical clutters of chromatic number three). New results are given for these clutters (mainly in corresponding gates synthesis). Even if older results have been proved and appeared in previous papers (from 1965 to 1983), we have choosen to recall all these proofs in order to make the paper self-contained.

Extending matchings in graphs: A survey

Gallai and Edmonds independently obtained a canonical decomposition of graphs in terms of their maximum matchings. Unfortunately, one of the degenerate cases for their theory occurs when the graph in question has a perfect matching (also known as a 1-factor). Kotzig, Lovász and others subsequently developed a further decomposition of such graphs. Among the ‘atoms’ of this decomposition is the family of bicritical graphs. (A graph G is bicritical if G - u - v has a perfect matching for every choice of two points u, v in G.) So far graphs have resisted further decomposition procedures.Motivated by these mysterious graphs, we introduced the following definition. Let p and n be positive integers with n </ (p − 2)/2. Graph G is n-extendable if G has a matching of size n and every such matching extends to (i.e. is a subset of) a perfect matching in G. It is clear that if a graph is bicritical, it is 1-extendable. A more interesting result is that if a graph is 2-extendable, it is either bipartite or bicritical. It is also true that if a graph is n-extendable, it is also (n − 1)-extendable. Hence, for nonbipartite graphs we have a nested sequence of families of bicritical graphs to study.In this paper, we survey a variety of results obtained over the past few years concerning n-extendable graphs. In particular, we describe how the property of n-extendability interacts with such other graph parameters as genus, toughness, claw-freedom and degree sums and generalized neighborhood conditions. We will also investigate the behavior of matching extendability under the operation of Cartesian product. The study of n-extendability for planar graphs has been, and continues to be, of particular interest.

Operations preserving the pfaffian property of a graph

AbstractPfaffian graphs are those which can be oriented so that the 1-factors have equal sign, as calculated according to the prescription of Kasteleyn. We consider various operations on graphs and examine their effect on the Pfaffian property. We show that the study of Pfaffian graphs may be reduced to the case of subcubic graphs (graphs in which no vertex has degree greater than 3) or bricks (3-connected bicritical graphs).

Matchings in regular graphs

We consider k-regular graphs with specified edge connectivity and show how some classical theorems and some new results concerning the existence of matchings in such graphs can be proved by using the polyhedral characterization of Edmonds. In addition, we show that lower bounds of Lovász and Plummer on the number of perfect matchings in bicritical graphs can be improved for cubic bicritical graphs.

On a Family of Planar Bicritical Graphs

Proceedings of the London Mathematical SocietyVolume s3-30, Issue 2 p. 160-176 Articles On a Family of Planar Bicritical Graphs L. Lovász, L. Lovász Eötvös L. University, Budapest, HungarySearch for more papers by this authorM. D. Plummer, M. D. Plummer Vanderbilt University, Nashville, TennesseeSearch for more papers by this author L. Lovász, L. Lovász Eötvös L. University, Budapest, HungarySearch for more papers by this authorM. D. Plummer, M. D. Plummer Vanderbilt University, Nashville, TennesseeSearch for more papers by this author First published: February 1975 https://doi.org/10.1112/plms/s3-30.2.160Citations: 24 † Work supported in part by NSF Contract GP-29129. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes3-30, Issue2February 1975Pages 160-176 RelatedInformation