Craig W. Rasmussen

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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) = NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors(More)
The competition graph of a loopless symmetric digraph If is the rwo-.\rc'p grclph. S,(H). Necessary and sufficient conditions on If are given for S,(ff) 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.(More)
Competition graphs were rst introduced by Joel Cohen in the study of food webs and have since been extensively studied. Graphs which are the competition graph of a strongly connected or Hamiltonian digraph are of particular interest in applications to communication networks. It has been previously established that every graph without isolated vertices(More)
The competition graph of a symmetric digraph D with a loop at each vertex is the square of the underlying graph of D with loops removed. In the interest of eeciently assigning radio frequencies in a communication network, we would like to determine which graphs have interval squares. Necessary and suucient conditions on the graph G are given for G 2 to be(More)
Large complex systems need to be analysed 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(More)
The p-competition graph G of a digraph D is a graph on the same vertex set as D, with x; y] 2 E(G) if and only if jOut(x) \Out(y)j p in D. In this paper we focus on the case in which D is a symmetric digraph ((a; b) is an arc in D if and only if (b; a) is an arc in D). We relate the problem to 2-step graphs, squares, and a generalization of the neighborhood(More)
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(More)
The (p)-neighborhood graph of a graph G, denoted N p (G), is de-ned on the same vertex set as G, with x; y] 2 E (N 2 (G)) if and only if jN(x)\N(y)j p in G, where N (v) is the open neighborhood of vertex v. The p]-neighborhood graph of G, N p G], is deened similarly, using closed neighborhoods rather than open ones. If G is the underlying graph of a(More)
Dedicated by the other authors to Professor John Maybee on the occasion of his 65th birthday. Abstract. 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(More)