Characterizations of competition multigraphs

AbstractThe notion of a competition multigraph was introduced by C. A. Anderson, K. F. Jones, J. R. Lundgren, and T. A. McKee [C. A. Anderson, K. F. Jones, J. R. Lundgren, and T. A. McKee: Competition multigraphs and the multicompetition number, Ars Combinatoria 29B (1990) 185–192] as a generalization of the competition graphs of digraphs.In this note, we give a characterization of competition multigraphs of arbitrary digraphs and a characterization of competition multigraphs of loopless digraphs. Moreover, we characterize multigraphs whose multicompetition numbers are at most m, where m is a given nonnegative integer and give characterizations of competition multihypergraphs

The double competition multigraph of a digraph

In this article, we introduce the notion of the double competition multigraph of a digraph. We give characterizations of the double competition multigraphs of arbitrary digraphs, loopless digraphs, reflexive digraphs, and acyclic digraphs in terms of edge clique partitions of the multigraphs.Comment: 9 page

Digraphs with Isomorphic Underlying and Domination Graphs: Pairs of Paths

A domination graph of a digraph D, dom (D), is created using thc vertex set of D and edge uv ϵ E (dom (D)) whenever (u, z) ϵ A (D) or (v, z) ϵ A (D) for any other vertex z ϵ A (D). Here, we consider directed graphs whose underlying graphs are isomorphic to their domination graphs. Specifically, digraphs are completely characterized where UGc (D) is the union of two disjoint paths

Ranking the nodes in directed and weighted directed graphs

Graphs;Econometrics

Competition graphs of degree bounded digraphs

If each vertex of an acyclic digraph has indegree at most $i$ and outdegree at most $j$, then it is called an $(i,j)$ digraph, which was introduced by Hefner~{\it et al.}~(1991). Whereas Hefner~{\it et al.} characterized $(i,j)$ digraphs whose competition graphs are interval, characterizing the competition graphs of $(i,j)$ digraphs is not an easy task. In this paper, we introduce the concept of $\langle i,j \rangle$ digraphs, which relax the acyclicity condition of $(i,j)$ digraphs, and study their competition graphs. By doing so, we obtain quite meaningful results. Firstly, we give a necessary and sufficient condition for a loopless graph being an $\langle i,j \rangle$ competition graph for some positive integers $i$ and $j$. Then we study on an $\langle i,j \rangle$ competition graph being chordal and present a forbidden subdigraph characterization. Finally, we study the family of $\langle i,j \rangle$ competition graphs, denoted by $\mathcal{G}_{\langle i,j \rangle}$, and identify the set containment relation on $\{\mathcal{G}_{\langle i,j \rangle}\colon\, i,j \ge 1\}$

Triangle-free mm-step competition graphs

Given a digraph $D$ and a positive number $m$, the $m$-step competition graph of $D$ is defined to have the same vertex set as $D$ and an edge joining two distinct vertices $u$ and $v$ if and only if there exist a $(u,w)$-directed walk and a $(v,w)$-directed walk both having length $m$ for some vertex $w$ in $D$. We call a graph an $m$-step competition graph if it is the $m$-step competition graph of a digraph. The notion of $m$-step competition graph was introduced by Cho \emph{et al.} \cite{cho2000m} as one of the variants of competition graph which was introduced by Cohen \cite{Cohen} while studying predator-prey concepts in ecological food webs. In this paper, we first extend a result given by Helleloid \cite{helleloid2005connected} stating that for all positive integers $m \geq n$, the only connected triangle-free $m$-step competition graph on $n$ vertices is the star graph. We show that the result is true for arbitrary positive integer $m \geq 2$ as long as it is the $m$-step competition graph of a digraph having a source. We go further to completely characterize the digraphs each of whose weak components has a source and whose $m$-step competition graphs are triangle-free for some integer $m \ge 2$. We also compute the number of digraphs with a source whose $m$-step competition graphs are connected and triangle-free for some integer $m \ge 2$