On chordal phylogeny graphs

An acyclic digraph each vertex of which has indegree at most $i$ and outdegree at most $j$ is called an $(i, j)$ digraph for some positive integers $i$ and $j$. Lee {\it et al.} (2017) studied the phylogeny graphs of $(2, 2)$ digraphs and gave sufficient conditions and necessary conditions for $(2, 2)$ digraphs having chordal phylogeny graphs. Their work was motivated by problems related to evidence propagation in a Bayesian network for which it is useful to know which acyclic digraphs have their moral graphs being chordal (phylogeny graphs are called moral graphs in Bayesian network theory). In this paper, we extend their work. We completely characterize phylogeny graphs of $(1, i)$ digraphs and $(i,1)$ digraphs, respectively, for a positive integer $i$. Then, we study phylogeny graphs of a $(2,j)$ digraphs, which is worthwhile in the context that a child has two biological parents in most species, to show that the phylogeny graph of a $(2,j)$ digraph $D$ is chordal if the underlying graph of $D$ is chordal for any positive integer $j$. Especially, we show that as long as the underlying graph of a $(2,2)$ digraph is chordal, its phylogeny graph is not only chordal but also planar.Comment: 18 page

On weak majority dimensions of digraphs

In this paper, we introduce the notion of the weak majority dimension of a digraph which is well-defined for any digraph. We first study properties shared by the weak dimension of a digraph and show that a weak majority dimension of a digraph can be arbitrarily large. Then we present a complete characterization of digraphs of weak majority dimension $0$ and $1$, respectively, and show that every digraph with weak majority dimension at most two is transitive. Finally, we compute the weak majority dimensions of directed paths and directed cycles and pose open problems

The phylogeny number in the aspect of triangles and diamonds of a graph

Given an acyclic digraph $D$, the competition graph of $D$, denoted by $C(D)$, is the simple graph having vertex set $V(D)$ and edge set $\{uv \mid (u, w), (v, w) \in A(D) \text{ for some } w \in V(D) \}$. The phylogeny graph of an acyclic digraph $D$, denoted by $P(D)$, is the graph with the vertex set $V(D)$ and the edge set $E(U(D)) \cup E(C(D))$ where $U(D)$ denotes the underlying graph of $D$. The notion of phylogeny graphs was introduced by Roberts and Sheng~\cite{roberts1997phylogeny} as a variant of competition graph. Moral graphs having arisen from studying Bayesian networks are the same as phylogeny graphs. In this paper, we integrate the existing theorems computing phylogeny numbers of connected graph with a small number of triangles into one proposition: for a graph $G$ containing at most two triangle, $|E(G)|-|V(G)|-2t(G)+d(G)+1 \le p(G) \le |E(G)|-|V(G)|-t(G)+1$ where $t(G)$ and $d(G)$ denote the number of triangles and the number of diamond in $G$, respectively. Then we show that these inequalities hold for graphs with many triangles. In the process of showing it, we derive a useful theorem which plays a key role in deducing various meaningful results including a theorem that answers a question given by Wu~{\it et al.}~\cite{Wu2019}

A new minimal chordal completion

In this paper, we present a minimal chordal completion $G^*$ of a graph $G$ satisfying the inequality $\omega(G^*) - \omega(G) \le i(G)$ for the non-chordality index $i(G)$ of $G$. In terms of our chordal completions, we partially settle the Hadwiger conjecture and the Erd\H{o}s-Faber-Lov\'{a}sz Conjecture, and extend the known $\chi$-bounded class by adding to it the family of graphs with bounded non-chordality indices

Competition numbers of planar graphs

In this paper, we relate the competition number of a graph to its edge clique cover number by presenting a tight inequality $k(G) \ge \theta_e(G)-|V(G)|+\widetilde{k}(G)$ where $\theta_e(G)$, $k(G)$, and $\widetilde{k}(G)$ are the edge clique cover number, the competition number, and the co-competition number of a graph $G$, respectively. By utilizing this inequality and a notion of competition-effective edge clique cover, we obtain some meaningful results on competition numbers of planar graphs.Comment: 15 pages, 3 figure

On the safe set of Cartesian product of two complete graphs

For a connected graph $G$, a vertex subset $S$ of $V(G)$ is a safe set if for every component $C$ of the subgraph of $G$ induced by $S$, $|C| \ge |D|$ holds for every component $D$ of $G-S$ such that there exists an edge between $C$ and $D$, and, in particular, if the subgraph induced by $S$ is connected, then $S$ is called a connected safe set. For a connected graph $G$, the safe number and the connected safe number of $G$ are the minimum among sizes of the safe sets and the minimum among sizes of the connected safe sets, respectively, of $G$. Fujita et al. introduced these notions in connection with a variation of the facility location problem. In this paper, we study the safe number and the connected safe number of Cartesian product of two complete graphs. Figuring out a way to reduce the number of components to two without changing the size of safe set makes it sufficient to consider only partitions of an integer into two parts without which it would be much more complicated to take care of all the partitions. In this way, we could show that the safe number and the connected safe number of Cartesian product of two complete graphs are equal and present a polynomial-time algorithm to compute them. Especially, in the case where one of complete components has order at most four, we precisely formulate those numbers.Comment: 15 pages, 8 figure

On mm-step competition graphs of bipartite tournaments

In this paper, we completely characterize the $m$-step competition graph of a bipartite tournament for any integer $m \ge 2$. In addition, we compute the competition index and the competition period of a bipartite tournament

Holes and a chordal cut in a graph

A set $X$ of vertices of a graph $G$ is called a {\em clique cut} of $G$ if the subgraph of $G$ induced by $X$ is a complete graph and the number of connected components of $G-X$ is greater than that of $G$. A clique cut $X$ of $G$ is called a {\em chordal cut} of $G$ if there exists a union $U$ of connected components of $G-X$ such that $G[U \cup X]$ is a chordal graph. In this paper, we consider the following problem: Given a graph $G$, does the graph have a chordal cut? We show that $K_{2,2,2}$-free hole-edge-disjoint graphs have chordal cuts if they satisfy a certain condition.Comment: 12 pages, 1 figur

The competition number of a graph in which any two holes share at most one edge

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs to characterize a graph by its competition number. A hole of a graph is a cycle of length at least 4 as an induced subgraph. It holds that the competition number of a graph cannot exceed one plus the number of its holes if G satisfies a certain condition. In this paper, we show that the competition number of a graph with exactly h holes any two of which share at most one edge is at most h+1, which generalizes the existing results on this subject.Comment: 29 pages, 14 figures, 1 tabl

On consecutive edge magic total labeling of connected bipartite graphs

Since Sedl\'{a}\breve{\mbox{c}}ek introduced the notion of magic labeling of a graph in 1963, a variety of magic labelings of a graph have been defined and studied. In this paper, we study consecutive edge magic labelings of a connected bipartite graph. We make a very useful observation that there are only four possible values of $b$ for which a connected bipartite graph has a $b$-edge consecutive magic labeling. On the basis of this fundamental result, we deduce various interesting results on consecutive edge magic labelings of bipartite graphs, especially caterpillars and lobsters, which extends the results given by Sugeng and Miller.Comment: 14 pages, 2 figure