David P. Sumner

Hamiltonian results in K1,3-free graphs

There 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.

Domination Critical Graphs

Abstract 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.

Every connected, locally connected nontrivial graph with no induced claw is hamiltonian

A 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.

Graphs with 1-factors

In this paper it is shown that if G is a connected graph of order 2n (n > 1) not containing a 1-factor, then for each k, 1 2. Let a, y E G be points of G which are a distance d apart and let D=a a ... xy be a path of length djoining a andy. Suppose Received by the editors September 5, 1972. AMS (MOS) subject classiications (1970). Primary 05C99. @ American Mathematical Society 1974

Longest paths and cycles in K1,3-free graphs

In 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.

Point determination in graphs

A point determining graph is defined to be a graph in which distinct nonadjacent points have distinct neighborhoods. Those graphs which are critical with respect to this property are studied. We show that a graph is complete if and only if it is connected, point determining, but fails to remain point determining upon the removal of any edge. We also show that every connected, point determining graph contains at least two points, the removal of either of which will result again in a point determining graph. Graphs which are point determining and contain exactly two such points are shown to have the property that every point is adjacent to exactly one of these two points.

Graphs indecomposable with respect to the X-join

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.

The diameter of domination k -critical graphs

A graph is k-domination-critical if γ(G) = k, and for any edge e not in G, γ(G + e) = k − 1. In this paper we show that the diameter of a domination k-critical graph with k ≧ 2 is at most 2k − 2. We also show that for every k ≧ 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.

Domination dot-critical graphs

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.

1-Factors of point determining graphs

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]: