On (1,2)(1,2)-step competition graphs of multipartite tournaments II

A multipartite tournament is an orientation of a complete $k$-partite graph for some positive integer $k\geq 3$. We say that a multipartite tournament $D$ is tight if every partite set forms a clique in the $(1,2)$-step competition graph, denoted by $C_{1,2}(D)$, of $D$. In the previous paper titled "On $(1,2)$-step competition graphs of multipartite tournaments" \cite{choi202412step} we completely characterize $C_{1,2}(D)$ for a tight multipartite tournament $D$. As an extension, in this paper, we study $(1,2)$-step competition graphs of multipartite tournaments that are not tight, which will be called loose. For a loose multipartite tournament $D$, various meaningful results are obtained in terms of $C_{1,2}(D)$ being interval and $C_{1,2}(D)$ being connected

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

A Complete Characterisation of Vertex-multiplications of Trees with Diameter 5

For a connected graph $G$, let $\mathscr{D}(G)$ be the family of strong orientations of $G$; and for any $D\in\mathscr{D}(G)$, we denote by $d(D)$ the diameter of $D$. The $\textit{orientation number}$ of $G$ is defined as $\bar{d}(G)=\min\{d(D)\mid D\in \mathscr{D}(G)\}$. In 2000, Koh and Tay introduced a new family of graphs, $G$ vertex-multiplications, and extended the results on the orientation number of complete $n$-partite graphs. Suppose $G$ has the vertex set $V(G)=\{v_1,v_2,\ldots, v_n\}$. For any sequence of $n$ positive integers $(s_i)$, a $G$ \textit{vertex-multiplication}, denoted by $G(s_1, s_2,\ldots, s_n)$, is the graph with vertex set $V^*=\bigcup_{i=1}^n{V_i}$ and edge set $E^*$, where $V_i$\u27s are pairwise disjoint sets with $|V_i|=s_i$, for $i=1,2,\ldots,n$; and for any $u,v\in V^*$, $uv\in E^*$ if and only if $u\in V_i$ and $v\in V_j$ for some $i,j\in \{1,2,\ldots, n\}$ with $i\neq j$ such that $v_i v_j\in E(G)$. They proved a fundamental classification of $G$ vertex-multiplications, with $s_i\ge 2$ for all $i=1,2,\ldots, n$, into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$, and any vertex-multiplication of a tree with diameter at least 3 does not belong to the class $\mathscr{C}_2$. Furthermore, some necessary and sufficient conditions for $\mathscr{C}_0$ were established for vertex-multiplications of trees with diameter $5$. In this paper, we give a complete characterisation of vertex-multiplications of trees with diameter $5$ in $\mathscr{C}_0$ and $\mathscr{C}_1$

Subquadratic-time algorithm for the diameter and all eccentricities on median graphs

On sparse graphs, Roditty and Williams [2013] proved that no $O(n^{2-\varepsilon})$-time algorithm achieves an approximation factor smaller than $\frac{3}{2}$ for the diameter problem unless SETH fails. In this article, we solve an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier? We propose the first combinatiorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for all eccentricities in median graphs with bounded dimension $d$, i.e. the dimension of the largest induced hypercube. This prerequisite on $d$ is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is $O(n^{1.6408}\log^{O(1)} n)$. We provide also some satellite outcomes related to this general result. In particular, restricted to simplex graphs, this algorithm enumerates all eccentricities with a quasilinear running time. Moreover, an algorithm is proposed to compute exactly all reach centralities in time $O(2^{3d}n\log^{O(1)}n)$.Comment: 43 pages, extended abstract in STACS 202