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Yoshio Sano

Publications and source records attributed to Yoshio Sano.

34 records · Page 2Linked to original sources
arXiv2012

The competition number of a generalized line graph is at most two

In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this note, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.

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arXiv2012

The competition hypergraphs of doubly partial orders

Since Cho and Kim (2005) showed that the competition graph of a doubly partial order is an interval graph, it has been actively studied whether or not the same phenomenon occurs for other variants of competition graph and interesting results have been obtained. Continuing in the same spirit, we study the competition hypergraph, an interesting variant of the competition graph, of a doubly partial order. Though it turns out that the competition hypergraph of a doubly partial order is not always interval, we completely characterize the competition hypergraphs of doubly partial orders which are interval.

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arXiv2012

A generalization of Opsut's lower bounds for the competition number of a graph

The notion of a competition graph was introduced by J. E. Cohen in 1968. The competition graph C(D) of a digraph $D$ is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices 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. In 1978, F. S. Roberts defined the competition number k(G) of a graph G as the minimum 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 the important research problems in the study of competition graphs to characterize a graph by its competition number. In 1982, R. J. Opsut gave two lower bounds for the competition number of a graph. In this paper, we give a generalization of these two lower bounds for the competition number of a graph.

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arXiv2011

The competition number of a graph and the dimension of its hole space

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. Recently, the relationship between the competition number and the number of holes of a graph is being studied. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In this paper, we conjecture that the dimension of the hole space of a graph is no smaller than the competition number of the graph. We verify this conjecture for various kinds of graphs and show that our conjectured inequality is indeed an equality for connected triangle-free graphs.

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arXiv2011

The competition numbers of ternary Hamming graphs

It is known to be a hard problem to compute the competition number k(G) of a graph G in general. Park and Sano [13] gave the exact values of the competition numbers of Hamming graphs H(n,q) if $1 \leq n \leq 3$ or $1 \leq q \leq 2$. In this paper, we give an explicit formula of the competition numbers of ternary Hamming graphs.

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arXiv2011

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.

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arXiv2011

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.

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arXiv2011

On the hypercompetition numbers of hypergraphs

The competition hypergraph $C{\cH}(D)$ of a digraph $D$ is the hypergraph such that the vertex set is the same as $D$ and $e \subseteq V(D)$ is a hyperedge if and only if $e$ contains at least 2 vertices and $e$ coincides with the in-neighborhood of some vertex $v$ in the digraph $D$. Any hypergraph with sufficiently many isolated vertices is the competition hypergraph of an acyclic digraph. The hypercompetition number $hk(\cH)$ of a hypergraph $\cH$ is defined to be the smallest number of such isolated vertices. In this paper, we study the hypercompetition numbers of hypergraphs. First, we give two lower bounds for the hypercompetition numbers which hold for any hypergraphs. And then, by using these results, we give the exact hypercompetition numbers for some family of uniform hypergraphs. In particular, we give the exact value of the hypercompetition number of a connected graph.

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arXiv2010

Graphs having many holes but with small competition numbers

The competition number k(G) of a graph G is the smallest number k such that G together with k isolated vertices added is the competition graph of an acyclic digraph. A chordless cycle of length at least 4 of a graph is called a hole of the graph. The number of holes of a graph is closely related to its competition number as the competition number of a graph which does not contain a hole is at most one and the competition number of a complete bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$ which has so many holes that no more holes can be added is the largest among those of graphs with n vertices. In this paper, we show that even if a connected graph G has many holes, the competition number of G can be as small as 2 under some assumption. In addition, we show that, for a connected graph G with exactly h holes and at most one non-edge maximal clique, if all the holes of G are pairwise edge-disjoint and the clique number $ω= ω(G)$ of G satisfies $2 \leq ω\leq h+1$, then the competition number of G is at most $h - ω+ 3$.

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arXiv2010

The competition numbers of Hamming graphs with diameter at most three

The competition graph of a digraph D is a 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 a graph G is defined to be 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. In this paper, we compute the competition numbers of Hamming graphs with diameter at most three.

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arXiv2010

The competition-common enemy graphs of digraphs satisfying Conditions $C(p)$ and $C'(p)$

S. -R. Kim and F. S. Roberts (2002) introduced the following conditions $C(p)$ and $C'(p)$ for digraphs as generalizations of the condition for digraphs to be semiorders. The condition $C(p)$ (resp. $C'(p)$) is: For any set $S$ of $p$ vertices in $D$, there exists $x \in S$ such that $N^+_D(x) \subseteq N^+_D(y)$ (resp. $N^-_D(x) \subseteq N^-_D(y)$) for all $y \in S$, where $N^+_D(x)$ (resp. $N^-_D(x)$) is the set of out-neighbors (resp. in-neighbors) of $x$ in $D$. The competition graph of a digraph $D$ is the (simple undirected) graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if $N^+_D(x) \cap N^+_D(y) \neq \emptyset$. Kim and Roberts characterized the competition graphs of digraphs which satisfy Condition $C(p)$. The competition-common enemy graph of a digraph $D$ is the graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if it holds that both $N^+_D(x) \cap N^+_D(y) \neq \emptyset$ and $N^-_D(x) \cap N^-_D(y) \neq \emptyset$. In this note, we characterize the competition-common enemy graphs of digraphs satisfying Conditions $C(p)$ and $C'(p)$.

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arXiv2009

The competition numbers of regular polyhedra

The notion of a competition graph was introduced by J. E. Cohen in 1968. The competition graph C(D) of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices 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 new isolated vertices is the competition graph of some acyclic digraph. In 1978, F. S. Roberts defined the competition number k(G) of a graph G as the minimum 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 computing its competition number. It is well known that there exist 5 kinds of regular polyhedra in the three dimensional Euclidean space: a tetrahedron, a hexahedron, an octahedron, a dodecahedron, and an icosahedron. We regard a polyhedron as a graph. The competition numbers of a tetrahedron, a hexahedron, an octahedron, and a dodecahedron are easily computed by using known results on competition numbers. In this paper, we focus on an icosahedron and give the exact value of the competition number of an icosahedron.

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arXiv2009

The competition number of a graph with exactly two holes

Let D be an acyclic digraph. The competition graph of D is a 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. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In 2005, Kim [5] conjectured that the competition number of a graph with h holes is at most h+1. Though Li and Chang [8] and Kim et al. [7] showed that her conjecture is true when the holes do not overlap much, it still remains open for the case where the holes share edges in an arbitrary way. In order to share an edge, a graph must have at least two holes and so it is natural to start with a graph with exactly two holes. In this paper, the conjecture is proved true for such a graph.

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arXiv2009

Cycles and p-competition graphs

The notion of p-competition graphs of digraphs was introduced by S-R. Kim, T. A. McKee, F. R. McMorris, and F. S. Roberts [p-competition graphs, Linear Algebra Appl., 217 (1995) 167--178] as a generalization of the competition graphs of digraphs. Let p be a positive integer. The p-competition graph C_p(D) of a digraph D=(V,A) is a (simple undirected) graph which has the same vertex set V and has an edge between distinct vertices x and y if and only if there exist p distinct vertices v_1, ..., v_p in V such that (x,v_i), (y,v_i) are arcs of the digraph D for each i=1, ..., p. In this paper, given a cycle of length n, we compute exact values of p in terms of n such that it is a p-competition graph, which generalizes the results obtained by Kim et al. We also find values of p in terms of n so that its complement is a p-competition graph.

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arXiv2009

The niche graphs of doubly partial orders

The competition graph of a doubly partial order is known to be an interval graph. The competition-common enemy graph of a doubly partial order is also known to be an interval graph unless it contains a cycle of length 4 as an induced subgraph. In this paper, we show that the niche graph of a doubly partial order is not necessarily an interval graph. In fact, we prove that, for each integer n at least 4, there exists a doubly partial order whose niche graph contains an induced subgraph isomorphic to a cycle of length n. We also show that if the niche graph of a doubly partial order is triangle-free, then it is an interval graph.

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