In this paper it is shown that if G is a connected graph of order 2 n ( n > 1 ) 2n(n > 1) not containing a 1-factor, then for each k, 1 > k ≦ n 1 > k \leqq n , there exists an induced; connected subgraph of order 2k which also fails to possess a 1-factor. Several other sufficient conditions for a graph to contain a 1-factor are presented. In particular, it is seen that the connected even order line graphs and total graphs always contain a 1-factor.

A graph is γ-excellent if every vertex of the graph is contained in some minimum dominating set of the graph. A vertex v is critical in G if the domination number of G-v is smaller than that of G. The graph G is dot-critical if contracting any edge of G produces a graph with smaller domination number. G is critically dominated if the set of critical vertices forms a dominating set for G. In this paper we show that these three properties, along with several others, are equivalent for trees on at least four vertices. We also provide a constructive characterization of these trees.

A graph G is dot-critical if contracting any edge decreases the domination number. It is totally dot-critical if identifying any two vertices decreases the domination number. We show that the totally dot-critical graphs essentially include the much-studied domination vertex-critical and edge-critical graphs as special cases. We investigate these properties, and provide a characterization of dot-critical and totally dot-critical graphs with domination number 2. We also consider the question of when a dot-critical graph contains a critical vertex.

A graph is k-domination-critical if gamma(G) = k, and for any edge e not in G, gamma(G + e) = k - 1. In this paper we show that the diameter of a domination k-critical graph with k greater-than-or-equal-to 2 is at most 2k - 2. We also show that for every k greater-than-or-equal-to 2, there is a k-domination-critical graph having diameter [(3/2)k - 1]. We also show that the diameter of a 4-domination-critical graph is at most 5. (C) 1994 John Wiley & Sons, Inc.

Publisher Summary Graphs that are minimal or critical with respect to a given property frequently play an important role in the investigation of that property. Not only are such graphs of considerable interest in their own right, but also a knowledge of their structure often aids in the development of the general theory. This chapter studies domination critical graphs that deals with those graphs that are critical in the sense that their domination number drops when any missing edge is added; and domination perfect graphs, is analogous to the idea of perfect graphs in the chromatic sense, and deals with those graphs that have all their induced subgraphs satisfying γ(G) = i(G) where i(G) is the independent domination number of G.

AbstractIn this article we show that the standard results concerning longest paths and cycles in graphs can be improved for K1,3‐free graphs. We obtain as a consequence of these results conditions for the existence of a hamiltonian path and cycle in K1,3‐free graphs.

AbstractThere have been a number of results dealing with Hamiltonian properties in powers of graphs. In this paper we show that the square and the total graph of a K1,3‐free graph are vertex pancyclic. We then discuss some of the relationships between connectivity and Hamiltonian properties in K1,3‐free graphs.

A set of vertices S is said to dominate the graph G if for each v ∉ S, there is a vertex u ∈ S with u adjacent to v. The smallest cardinality of any such dominating set is called the domination number of G and is denoted by γ(G). The purpose of this paper is to initiate an investigation of those graphs which are critical in the following sense: For each v, u ∈ V(G) with v not adjacent to u, γ(G + vu) < γ(G). Thus G is k-y-critical if γ(G) = k and for each edge e ∉ E(G), γ(G + e) = k −1. The 2-domination critical graphs are characterized the properties of the k-critical graphs with k ≥ 3 are studied. In particular, the connected 3-critical graphs of even order are shown to have a 1-factor and some stringent restrictions on their degree sequences and diameters are obtained.

AbstractA graph is defined to be randomly matchable if every matching of G can be extended to a perfect matching. It is shown that the connected randomly matchable graphs are precisely K2n and Kn,n (n ≥ 1).

AbstractA graph is locally connected if every neighborthood induces a connected subgraph. We show here that every connected, locally connected graph on p ≥ 3 vertices and having no induced K1,3 is Hamiltonian. Several sufficient conditions for a line graph to be Hamiltonian are obtained as corollaries.

AbstractA graph is point determining if distinct vertices have distinct neighborhoods. The nucleus of a point‐determining graph is the set GO of all vertices, v, such that G–v is point determining. In this paper we show that the size, ω(G), of a maximum clique in G satisfies ω(G) ⩽ 2|π (G)O|, where π(G) (the point determinant of G) is obtained from G by identifying vertices which have the same neighborhood.

A graph G is totally connected if both G and Ḡ (its complement) are connected. The connected Ramsey number r c ( F , H ) is the smallest integer k ⩾ 4 so that if G is a totally connected graph of order k then either F ⊂ G or H ⊂ Ḡ . We show that if neither of F nor H contains a bridge, then r c = r ( F , H ), the usual generalized Ramsey number of F and H . We compute r c ( P m P m ), the connected Ramsey number for paths.

In this paper we investigate the edge nucleus E0(G) of a point-determining graph G. We observe several relationships between E0(G) and the nucleus G0 = {v ∈ V(G)∣ G − v is point determining} and use these relationships to prove several properties of E0(G). In particular, we show that there are only a finite number of graphs with a given edge nucleus and we determine those graphs G for which |E0(G)| ≤ 2. We also show that an n-clique of a point-determining graph G contains at least n−2 edges of E0(G) and if G is totally point determining, then every odd cycle of G meets E0(G).

Dushnik and Miller defined the dimension of a partial order P as the minimum number of linear orders whose intersection is P. Ken Bogart asked if the dimension of a partial order is an invariant of the associated comparability graph.In this paper we answer Bogart's question in the affirmative.The proof involves a characterization of the class of comparability graphs defined by Aigner and Prins as uniquely partially orderable graphs.Our characterization of uniquely partially orderable graphs is another instance of the frequently encountered phenomenon where the obvious necessary condition is also sufficient.

Journal of the London Mathematical SocietyVolume s2-13, Issue 2 p. 351-359 Notes and Papers 1-Factors and Antifactor Sets David P. Sumner, David P. Sumner University of South Carolina, Columbia, South Carolina 29208Search for more papers by this author David P. Sumner, David P. Sumner University of South Carolina, Columbia, South Carolina 29208Search for more papers by this author First published: June 1976 https://doi.org/10.1112/jlms/s2-13.2.351Citations: 46AboutPDF 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 Volumes2-13, Issue2June 1976Pages 351-359 RelatedInformation

In this paper all graphs will be ordinary graphs, i.e. finite, undirected, and without loops or multiple edges. For points x and y of a graph G, we shall indicate that x is adjacent to y by writing x ⊥ y, and if x is not adjacent to y we shall write xy. We shall denote the degree of a point x by δ(x) and the minimal degree of G by δ(G).By the line graph of a graph G we shall mean the graph L(G) whose points are the edges of G, with two points of L(G) adjacent whenever they are adjacent in G. A graph G is said to be a line graph if there exists a graph H such that G = L(H).

In this paper all graphs will be finite, undirected, and without loops or multiple edges. We continue the investigation initiated in [l] and obtain new results concerning point determining graphs and some applications of these results to the theory of l-factors. Our notation and terminology will conform to that in [I]; in particular, we will consider a graph G to consist of a set of points (which we also denote by G) together with an adjacency relation I, i.e., LI i b if and only if a and b are adjacent points of G. If a and b are not adjacent, we will write a J b. If A C G, A’ = {x / x 1 a for all a E A}; however, instead of {xl’ we write just XI. If a 1 b, we denote the edge with endpoints a and b as ub. A graph G is point determining if and only if for every two points a, b E G with a # b we have ui :+ b-l, i.e., 0 and b have distinct neighborhoods. If G is any graph, then the point determining graph obtained from G by identifying a with b whenever ui == bL is called thepoinr determinant of G and will be denoted by n(G). When no confusion is possible, we will not distinguish between a subset of the points A _C G and the subgraph that it induces. We will denote the cardinality of a set A bylA/. For completeness we include the following results (Lemmas 1-3) whose proofs may be found in [I]:

In this paper our graphs will be finite, undirected, and without loops or multiple edges. We will denote the set of vertices of a graph G by V(G). If G is a graph and u, v∈V(G), then we will write u ∼ v to denote that u and v are adjacent and u ≁ v otherwise. If A ⊆ V(G), then we let N(A) = {u∈ V(G)|u ∼ a for each a ∈A}. However we write N(v) instead of N({v}). When there is no chance of confusion, we will not distinguish between a subset A ⊆ V(G) of vertices of G and the subgraph that it induces. We will denote the cardinality of a set A by |A|. The degree of a vertex v is δ(v) = |N(v)|. Any undefined terminology in this paper will generally conform with Behzad and Chartrand [1].

In his paper [3], Sabidussi defined the X-join of a family of graphs. This concept has also appeared in the work of Foulis and Randall on empirical logic [1,2]. In this paper, we investigate those graphs which do not have a nontrivial representation as the X-join of some family of graphs.