Ronald D. Dutton

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.

Weak-heap sort

A data structure called aweak-heap is defined by relaxing the requirements for a heap. The structure can be implemented on a 1-dimensional array with one extra bit per data item and can be initialized withn items using exactlyn−1 data element compares. Theoretical analysis and empirical results indicate that it is a competitive structure for sorting. The worst case number of data element comparisons is strictly less than (n−1) logn+0.086013n and the expected number is conjectured to be approximately (n−0.5)logn−0.413n.

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.

The Optimal Location of Nuclear-Power Facilities in the Pacific Northwest

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.

A compilation of relations between graph invariants—supplement I

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.