Robert C. Brigham

Vertex domination‐critical graphs

A dominating set in a graph G is a set of vertices D such that every vertex of G is either in D or is adjacent some vertex of D. The domination number Γ(G) of G is the minimum cardinality of any dominating set. A graph is vertex domination-critical if the removal of any vertex decreases its domination number. This paper gives examples and properties of vertex domination-critical graphs, presents a method of constructing them, and poses some open questions. In the process several results for arbitrary graphs are presented.

A characterization of competition graphs

Abstract Characterization of competition graphs for arbitrary and acyclic directed graphs are presented.

Edges in graphs with large girth

AbstractSeveral upper bounds are given for the maximum number of edgese possible in a graph depending upon its orderp, girthg and, in certain cases, minimum degreeδ. In particular, one upper bound has an asymptotic order ofp1+2/(g−1) wheng is odd. A corollary of our final result is that

Powerful alliances in graphs

g \leqslant 2 + 2\log _k \left( {\frac{p}{4}} \right)

An external problem for edge domination insensitive graphs

whenk = ⌊e/p⌋ ≥ 2. Asymptotic and numerical comparisons are also presented.

A compilation of relations between graph invariants—supplement I

Abstract Given a factoring of a graph, the factor domination number γ f is the smallest number of nodes which dominate all factors. General results, mainly involving bounds on γ f for factoring of arbitrary graphs, are presented, and some of these are generalizations of well known relationships. The special case of two-factoring K p into a graph G and its complement G receives special emphasis.

Bicritical domination

For a graph G=(V,E), a non-empty set S@?V is a defensive alliance if for every vertex v in S, v has at most one more neighbor in V-S than it has in S, and S is an offensive alliance if for every v@?V-S that has a neighbor in S, v has more neighbors in S than in V-S. A powerful alliance is both defensive and offensive. We initiate the study of powerful alliances in graphs.

INGRID: A graph invariant manipulator

Let G=(V,E) be a graph. A set S@?V is a defensive alliance if |N[x]@?S|>=|N[x]-S| for every x@?S. Thus, each vertex of a defensive alliance can, with the aid of its neighbors in S, be defended from attack by its neighbors outside of S. An entire set S is secure if any subset X@?S can be defended from an attack from outside of S, under an appropriate definition of what such a defense implies. Necessary and sufficient conditions for a set to be secure are determined.