Competitively tight graphs

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 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. The competition number $k(G)$ of a graph $G$ is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph $G$ is related to the edge clique cover number $\theta_E(G)$ of the graph $G$ via $\theta_E(G)-|V(G)|+2 \leq k(G) \leq \theta_E(G)$. We first show that for any positive integer $m$ satisfying $2 \leq m \leq |V(G)|$, there exists a graph $G$ with $k(G)=\theta_E(G)-|V(G)|+m$ and characterize a graph $G$ satisfying $k(G)=\theta_E(G)$. We then focus on what we call \emph{competitively tight graphs} $G$ which satisfy the lower bound, i.e., $k(G)=\theta_E(G)-|V(G)|+2$. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.Comment: 10 pages, 2 figure

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.Comment: 6 pages, 2 figure

생태계에서의 경쟁 관점으로 그래프와 유향그래프의 구조 연구

학위논문(박사) -- 서울대학교대학원 : 사범대학 수학교육과, 2023. 2. 김서령.In this thesis, we study m-step competition graphs, (1, 2)-step competition graphs, phylogeny graphs, and competition-common enemy graphs (CCE graphs), which are primary variants of competition graphs. Cohen [11] introduced the notion of competition graph while studying predator-prey concepts in ecological food webs.An ecosystem is a biological community of interacting species and their physical environment. For each species in an ecosystem, there can be m conditions of the good environment by regarding lower and upper bounds on numerous dimensions such as soil, climate, temperature, etc, which may be represented by an m-dimensional rectangle, so-called an ecological niche. An elemental ecological truth is that two species compete if and only if their ecological niches overlap. Biologists often describe competitive relations among species cohabiting in a community by a food web that is a digraph whose vertices are the species and an arc goes from a predator to a prey. In this context, Cohen [11] defined the competition graph of a digraph as follows. The competition graph C(D) of a digraph D is defined to be a simple graph whose vertex set is the same as V (D) and which has an edge joining two distinct vertices u and v if and only if there are arcs (u, w) and (v, w) for some vertex w in D. Since Cohen introduced this definition, its variants such as m-step competition graphs, (i, j)-step competition graphs, phylogeny graphs, CCE graphs, p-competition graphs, and niche graphs have been introduced and studied. As part of these studies, we show that the connected triangle-free m-step competition graph on n vertices is a tree and completely characterize the digraphs of order n whose m-step competition graphs are star graphs for positive integers 2 ≤ m < n. We completely identify (1,2)-step competition graphs C_{1,2}(D) of orientations D of a complete k-partite graph for some k ≥ 3 when each partite set of D forms a clique in C_{1,2}(D). In addition, we show that the diameter of each component of C_{1,2}(D) is at most three and provide a sharp upper bound on the domination number of C_{1,2}(D) and give a sufficient condition for C_{1,2}(D) being an interval graph. On the other hand, we study on phylogeny graphs and CCE graphs of degreebounded acyclic digraphs. An acyclic digraph in which every vertex has indegree at most i and outdegree at most j is called an (i, j) digraph for some positive integers i and j. If each vertex of a (not necessarily acyclic) digraph D has indegree at most i and outdegree at most j, then D is called an hi, ji digraph. We give a sufficient condition on the size of hole of an underlying graph of an (i, 2) digraph D for the phylogeny graph of D being a chordal graph where D is an (i, 2) digraph. Moreover, we go further to completely characterize phylogeny graphs of (i, j) digraphs by listing the forbidden induced subgraphs. We completely identify the graphs with the least components among the CCE graphs of (2, 2) digraphs containing at most one cycle and exactly two isolated vertices, and their digraphs. Finally, we gives a sufficient condition for CCE graphs being interval graphs.이 논문에서 경쟁그래프의 주요 변이들 중 m-step 경쟁그래프, (1, 2)-step 경쟁 그래프, 계통 그래프, 경쟁공적그래프에 대한 연구 결과를 종합했다. Cohen [11]은 먹이사슬에서 포식자-피식자 개념을 연구하면서 경쟁그래프 개념을 고안했다. 생태계는 상호작용하는 종들과 그들의 물리적 환경의 생물학적 체계이다. 생태계의 각 종에 대해서, 토양, 기후, 온도 등과 같은 다양한 차원의 하계 및 상계를 고려하여 좋은 환경을 m개의 조건들로 나타낼 수 있는데 이를 생태적 지위(ecological niche)라고 한다. 생태학적 기본가정은 두 종이 생태적 지위가 겹치면 경쟁하고(compete), 경쟁하는 두 종은 생태적 지위가 겹친다는 것이다. 흔히 생물학자들은 한 체제에서 서식하는 종들의 경쟁적 관계를 각 종은 꼭짓점으로, 포식자에서 피식자에게는 유향변(arc)을 그어서 먹이사슬로 표현한다. 이러한 맥락에서 Cohen [11]은 다음과 같이 유향그래프의 경쟁 그래프를 정의했다. 유향그래프(digraph) D의 경쟁그래프(competition graph) C(D) 란 V (D)를 꼭짓점 집합으로 하고 두 꼭짓점 u, v를 양 끝점으로 갖는 변이 존재한다는 것과 꼭짓점 w가 존재하여 (u, w),(v, w)가 모두 D에서 유향변이 되는 것이 동치인 그래프를 의미한다. Cohen이 경쟁그래프의 정의를 도입한 이후로 그 변이들로 m-step 경쟁그래프(m-step competition graph), (i, j)-step 경쟁그래프((i, j)-step competition graph), 계통그래프(phylogeny graph), 경쟁공적그래프(competition-common enemy graph), p-경쟁그래프(p-competition graph), 그리고 지위그래프(niche graph)가 도입되었고 연구되고 있다. 이 논문의 연구 결과들의 일부는 다음과 같다. 삼각형이 없이 연결된 m-step 경쟁 그래프는 트리(tree)임을 보였으며 2 ≤ m < n을 만족하는 정수 m, n에 대하여 꼭짓점의 개수가 n개이고 m-step 경쟁그래프가 별그래프(star graph)가 되는 유향그래프를 완벽하게 특징화 하였다. k ≥ 3이고 방향지어진 완전 k-분할 그래프(oriented complete k-partite graph)의 (1, 2)-step 경쟁그래프 C_{1,2}(D)에서 각 분할이 완전 부분 그래프를 이룰 때, C_{1,2}(D)을 모두 특징화 하였다. 또한, C_{1,2}(D)의 각 성분(component)의 지름(diameter)의 길이가 최대 3이며 C_{1,2}(D)의 지배수(domination number)에 대한 상계와 최댓값을 구하고 구간그래프(interval graph)가 되기 위한 충분 조건을 구하였다. 차수가 제한된 유향회로를 갖지 않는 유향그래프(degree-bounded acyclic digraph)의 계통그래프와 경쟁공적그래프에 대해서도 연구하였다. 양의 정수들 i, j에 대하여 (i, j) 유향그래프란 각 꼭짓점의 내차수는 최대 i, 외차수는 최대 j인 유향회로 갖지 않는 유향그래프이다. 만약 유향그래프 D에 각 꼭짓점이 내차수가 최대 i, 외차수가 최대 j 인 경우에 D를 hi, ji 유향그래프라 한다. D가 (i, 2) 유향그래프일 때, D의 계통그래프가 현그래프(chordal graph)가 되기 위한 D의 방향을 고려하지 않고 얻어지는 그래프(underlying graph)에서 길이가 4이상인 회로(hole)의 길이에 대한 충분조건을 구하였다. 게다가 (i, j) 유향그래프의 계통그래프에서 나올 수 없는 생성 부분 그래프(forbidden induced subgraph)를 특징화 하였다. (2, 2) 유향그래프 D의 경쟁공적그래프 CCE(D)가 2개의 고립점(isolated vertex)과 최대 1개의 회로를 갖으면서 가장 적은 성분을 갖는 경우일 때의 구조를 규명했다. 마지막으로, CCE(D)가 구간그래프가 되기 위한 성분의 개수에 대한 충분조건을 구하였다.1 Introduction 1 1.1 Graph theory terminology and basic concepts 1 1.2 Competition graphs and its variants 6 1.2.1 A brief background of competition graphs 6 1.2.2 Variants of competition graphs 8 1.2.3 m-step competition graphs 10 1.2.4 (1, 2)-step competition graphs 13 1.2.5 Phylogeny graphs 14 1.2.6 CCE graphs 16 1.3 A preview of the thesis 17 2 Digraphs whose m-step competition graphs are trees 19 2.1 The triangle-free m-step competition graphs 23 2.2 Digraphs whose m-step competition graphs are trees 29 2.3 The digraphs whose m-step competition graphs are star graphs 38 3 On (1, 2)-step competition graphs of multipartite tournaments 47 3.1 Preliminaries 48 3.2 C1,2(D) with a non-clique partite set of D 51 3.3 C1,2(D) without a non-clique partite set of D 66 3.4 C1,2(D) as a complete graph 74 3.5 Diameters and domination numbers of C1,2(D) 79 3.6 Disconnected (1, 2)-step competition graphs 82 3.7 Interval (1, 2)-step competition graphs 84 4 The forbidden induced subgraphs of (i, j) phylogeny graphs 90 4.1 A necessary condition for an (i, 2) phylogeny graph being chordal 91 4.2 Forbidden subgraphs for phylogeny graphs of degree bounded digraphs 99 5 On CCE graphs of (2, 2) digraphs 122 5.1 CCE graphs of h2, 2i digraphs 128 5.2 CCE graphs of (2, 2) digraphs 134 Abstract (in Korean) 168 Acknowledgement (in Korean) 170박

The (1,2)-Step Competition Graph of a Tournament

The competition graph of a digraph, introduced by Cohen in 1968, has been extensively studied. More recently, in 2000, Cho, Kim, and Nam defined the m-step competition graph. In this paper, we offer another generalization of the competition graph. We define the (1,2)-step competition graph of a digraph D, denoted C1,2(D), as the graph on V(D) where {x,y}∈E(C1,2(D)) if and only if there exists a vertex z≠x,y, such that either dD−y(x,z)=1 and dD−x(y,z)≤2 or dD−x(y,z)=1 and dD−y(x,z)≤2. In this paper, we characterize the (1,2)-step competition graphs of tournaments and extend our results to the (i,k)-step competition graph of a tournament

On p-competition graphs of loopless Hamiltonian digraphs without symmetric arcs and graph operations

For a digraph $D$, the $p$-competition graph $C_{p}(D)$ of $D$ is the graph satisfying the following: $V(C_{p}(D))=V(D)$, for $x,y \in V(C_{p}(D))$, $xy \in E(C_{p}(D))$ if and only if there exist distinct $p$ vertices $v_{1},$ $v_{2},$ $...,$ $v_{p}$ $\in$ $V(D)$ such that $x \rightarrow v_{i}$, $y \rightarrow v_{i}$ $\in$ $A(D)$ for each $i=1,2,$ $...,$ $p$. We show the $H_1 \cup H_2$ is a $p$-competition graph of a loopless digraph without symmetric arcs for $p \geq 2$, where $H_1$ and $H_2$ are $p$-competition graphs of loopless digraphs without symmetric arcs and $V(H_1) \cap V(H_2)$ $=$ $\{ \alpha \}$. For $p$-competition graphs of loopless Hamiltonian digraphs without symmetric arcs, we obtain similar results. And we show that a star $K_{1,n}$ is a $p$-competition graph of a loopless Hamiltonian digraph without symmetric arcs if $n \geq 2p+3$ and $p \geq 3$. Based on these results, we obtain conditions such that spiders, caterpillars and cacti are $p$-competition graphs of loopless digraphs without symmetric arcs. We also obtain conditions such that these graphs are $p$-competition graphs of loopless Hamiltonian digraphs without symmetric arcs

The Competition Numbers of Johnson Graphs with Diameter Four

In 2010, Kim, Park and Sano studied the competition numbers of Johnson graphs. They gave the competition numbers of J(n,2) and J(n,3).In this note, we consider the competition number of J(n,4)

d-반순서의 경쟁그래프의 연구

학위논문 (박사)-- 서울대학교 대학원 : 사범대학 수학교육과, 2018. 2. 김서령.The \emph{competition graph} $C(D)$ of a digraph $D$ is defined to be a graph whose vertex set is the same as $D$ and which has an edge joining two distinct vertices $x$ and $y$ if and only if there are arcs $(x,z)$ and $(y,z)$ for some vertex $z$ in $D$. Competition graphs have been extensively studied for more than four decades. Cohen~\cite{cohen1968interval, cohen1977food, cohen1978food} empirically observed that most competition graphs of acyclic digraphs representing food webs are interval graphs. Roberts~\cite{roberts1978food} asked whether or not Cohen's observation was just an artifact of the construction, and then concluded that it was not by showing that if $G$ is an arbitrary graph, then $G$ together with additional isolated vertices as many as the number of edges of $G$ is the competition graph of some acyclic digraph. Then he asked for a characterization of acyclic digraphs whose competition graphs are interval graphs. Since then, the problem has remained elusive and it has been one of the basic open problems in the study of competition graphs. There have been a lot of efforts to settle the problem and some progress has been made. While Cho and Kim~\cite{cho2005class} tried to answer his question, they could show that the competition graphs of doubly partial orders are interval graphs. They also showed that an interval graph together with sufficiently many isolated vertices is the competition graph of a doubly partial order. In this thesis, we study the competition graphs of $d$-partial orders some of which generalize the results on the competition graphs of doubly partial orders. For a positive integer $d$, a digraph $D$ is called a \emph{$d$-partial order} if V(D) \subset \RR^d and there is an arc from a vertex $\mathbf{x}$ to a vertex $\mathbf{y}$ if and only if $\mathbf{x}$ is componentwise greater than $\mathbf{y}$. A doubly partial order is a $2$-partial order. We show that every graph $G$ is the competition graph of a $d$-partial order for some nonnegative integer $d$, call the smallest such $d$ the \emph{partial order competition dimension} of $G$, and denote it by $\dim_\text{poc}(G)$. This notion extends the statement that the competition graph of a doubly partial order is interval and the statement that any interval graph can be the competition graph of a doubly partial order as long as sufficiently many isolated vertices are added, which were proven by Cho and Kim~\cite{cho2005class}. Then we study the partial order competition dimensions of some interesting families of graphs. We also study the $m$-step competition graphs and the competition hypergraph of $d$-partial orders.1 Introduction 1 1.1 Basic notions in graph theory 1 1.2 Competition graphs 6 1.2.1 A brief history of competition graphs 6 1.2.2 Competition numbers 7 1.2.3 Interval competition graphs 10 1.3 Variants of competition graphs 14 1.3.1 m-step competition graphs 15 1.3.2 Competition hypergraphs 16 1.4 A preview of the thesis 18 2 On the competition graphs of d-partial orders 1 20 2.1 The notion of d-partial order 20 2.2 The competition graphs of d-partial orders 21 2.2.1 The regular (d − 1)-dimensional simplex △ d−1 (p) 22 2.2.2 A bijection from H d + to a set of regular (d − 1)-simplices 23 2.2.3 A characterization of the competition graphs of d-partial orders 25 2.2.4 Intersection graphs and competition graphs of d-partial orders 27 2.3 The partial order competition dimension of a graph 29 3 On the partial order competition dimensions of chordal graphs 2 38 3.1 Basic properties on the competition graphs of 3-partial orders 39 3.2 The partial order competition dimensions of diamond-free chordal graphs 42 3.3 Chordal graphs having partial order competition dimension greater than three 46 4 The partial order competition dimensions of bipartite graphs 3 53 4.1 Order types of two points in R 3 53 4.2 An upper bound for the the partial order competition dimension of a graph 57 4.3 Partial order competition dimensions of bipartite graphs 64 5 On the m-step competition graphs of d-partial orders 4 69 5.1 A characterization of the m-step competition graphs of dpartial orders 69 5.2 Partial order m-step competition dimensions of graphs 71 5.3 dim poc (Gm) in the aspect of dim poc (G) 76 5.4 Partial order competition exponents of graphs 79 6 On the competition hypergraphs of d-partial orders 5 81 6.1 A characterization of the competition hypergraphs of d-partial orders 81 6.2 The partial order competition hyper-dimension of a hypergraph 82 6.3 Interval competition hypergraphs 88 Abstract (in Korean) 99Docto

방향 지어진 완전 이분 그래프의 m-step 경쟁 그래프

학위논문 (석사)-- 서울대학교 대학원 : 사범대학 수학교육과, 2018. 2. 김서령.In this thesis, we study the m-step competition graphs of bipartite tournaments. We compute the number of distinct bipartite tournaments by Polya's theory of counting. Then we study the competition indices and competition periods of bipartite tournaments. We characterize the pairs of graphs that can be represented as the m-step competition graphs of bipartite tournaments. Finally, we present the maximum number of edges and the minimum number of edges which the m-step competition graph of a bipartite tournament might have.1 Introduction 1 1.1 Basic graph terminology 1 1.2 Competition graph and its variants 3 1.3 m-step competition graphs 5 1.4 Polya's theory of counting 6 1.5 Competition indices and competition periods 9 1.6 Preview of thesis 10 2 The number of distinct bipartite tournaments 12 3 Properties of m-step competition graphs of bipartite tournaments 18 4 m-step competition realizable pairs 23 5 Extremal cases 31 6 Concluding remarks 36Maste

Minimizing Direct Competitions in Complete Components of (1,2)-Step Competition Graphs

Graph theory is a useful tool for studying systems of food webs, a concept from ecology that models the predator-prey relationships between species in an ecosystem. We have used this concept to inform and motivate our exploration of graph theory. In particular we examine the characteristics of (1,2)-step competition graphs developed by Factor and Merz in 2010, which are an extension of normal competition graphs first introduced by Cohen in 1968. Factor and Merz define the (1,2)-step competition graph of a digraph D, denoted by C1,2(D), as the graph with the same vertex set as D and an edge between vertices x and y if and only if there exists some z in V(D) for which either dD\{x}(y, z) = 1 and dD\{y}(x, z) =\u3c 2 or, dD\{y}(x, z) = 1 and dD\{x}(y, z) =\u3c 2. We extend this definition and say that given x, y in V(D) such that (x, y) in E(C1,2), x and y compete directly if there exists a vertex z in D such that d(y, z) = 1 and d(x, z) = 1. We then call the edge (x, y) in E(C1,2) a direct competition between x and y. Otherwise, we say that x and y compete indirectly and we call the edge (x, y) in E(C1,2) an indirect competition between x and y. We have developed a family of digraphs that induce complete components in their (1,2)-step competition graphs that appear to have a minimum number of direct competitions