Craig W. Rasmussen

Interval competition graphs of symmetric digraphs

Abstract The competition graph of a loopless symmetric digraph H is the two-step graph , S 2 ( H ). Necessary and sufficient conditions on H are given for S 2 ( H ) to be interval or unit interval. These are useful properties when application requires that the competition graph be efficiently colorable. Computational aspects are discussed, as are related open problems.

Two-step graphs of trees

Abstract In this paper we examine the two-step graphs of trees in two distinct contexts. We first consider the two-step graph S 2 ( T ) of a tree T as the competition graph of T and give necessary and sufficient conditions for the competition graph of a tree to be an interval graph. We then investigate the problem of inverting the two-step S 2 ( T ) of a tree T , i.e., reconstructing T from S 2 ( T ). A class of trees is identified whose two-step graphs are invertible.

Chromatic numbers of competition graphs

Abstract : Previous work on competition graphs has emphasized characterization, not only of the competition graphs themselves but also of those graphs whose competition graphs are chordal or interval. The latter sort of characterization is of interest when a competition graph that is easily colorable would be useful, e.g. in a scheduling or assignment problem. This leads naturally to the following question: Given a graph F, does the structure of G tell us anything about the chromatic number X of the competition graph C(G)? We show that in some cases we can calculate this chromatic number exactly, while in others we can place tight bounds on the chromatic number.

The set chromatic number of a graph

For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G, the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) 6= NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors required of such a coloring is called the set chromatic number χs(G) of G. The set chromatic numbers of some well-known classes of graphs are determined and several bounds are established for the set chromatic number of a graph in terms of other graphical parameters.

A Characterization of Graphs With Interval Two-Step Graphs

One of the intriguing open problems on competition graphs is determining what digraphs have interval competition graphs. This problem originated in the work of Cohen [5,6] on food webs. In this paper we consider this problem for the class of loopless symmetric digraphs. The competition graph of a symmetric digraph D is the two-step graph of the underlying graph H of D, denoted S_2(H). The two-step graph is also known as the neighborhood graph, and has been studied recently by Brigham and Dutton [4] and Boland, Brigham and DUTTON [1,2]. This work was motivated by a paper of Raychaudhuri and Roberts [20] where they investigated symmetric digraphs with a loop at each vertex. Under these assumption, the competition graph is the square of the underlying graph H without loops. Here we will first consider forbidden subgraph characterizations of graphs with interval two-step graphs. Second, we will characterize a large class of graphs with interval two-step graphs using the Gilmore-Hoffman characterization of interval graphs.

A program for education in certification and accreditation

Large complex systems need to be analyzed prior to operation so that those depending upon them for the protection of their information have a well-defined understanding of the measures that have been taken to achieve security and the residual risk the system owner assumes during its operation. The U.S. military calls this analysis and vetting process certification and accreditation. Today there is a large, unsatisfied need for personnel qualified to conduct system certifications. An educational program to address those needs is described.