Recognizing sparse perfect elimination bipartite graphs

When applying Gaussian elimination to a sparse matrix, it is desirable to avoid turning zeros into non-zeros to preserve the sparsity. The class of perfect elimination bipartite graphs is closely related to square matrices that Gaussian elimination can be applied to without turning any zero into a non-zero. Existing literature on the recognition of this class and finding suitable pivots mainly focusses on time complexity. For $n \times n$ matrices with m non-zero elements, the currently best known algorithm has a time complexity of $O(n^3/\log n)$. However, when viewed from a practical perspective, the space complexity also deserves attention: it may not be worthwhile to look for a suitable set of pivots for a sparse matrix if this requires $\Omega(n^2)$ space. We present two new algorithms for the recognition of sparse instances: one with a $O(n m)$ time complexity in $\Theta(n^2)$ space and one with a $O(m^2)$ time complexity in $\Theta(m)$ space. Furthermore, if we allow only pivots on the diagonal, our second algorithm can easily be adapted to run in time $O(n m)$

Semi-strict chordality of digraphs

Chordal graphs are important in algorithmic graph theory. Chordal digraphs are a digraph analogue of chordal graphs and have been a subject of active studies recently. In this paper we introduce the notion of semi-strict chordal digraphs which form a class strictly between chordal digraphs and chordal graphs. We characterize semi-strict chordal digraphs by forbidden subdigraphs within the cases of locally semicomplete digraphs and weakly quasi-transitive digraphs.Comment: 12 pages, 2 figures. arXiv admin note: text overlap with arXiv:2008.0356

Permanents, Pfaffian orientations, and even directed circuits

Given a 0-1 square matrix A, when can some of the 1's be changed to -1's in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign or are both zero) is nonsingular? When is a hypergraph with n vertices and n hyperedges minimally nonbipartite? When does a bipartite graph have a "Pfaffian orientation"? Given a digraph, does it have no directed circuit of even length? Given a digraph, does it have a subdivision with no even directed circuit? It is known that all of the above problems are equivalent. We prove a structural characterization of the feasible instances, which implies a polynomial-time algorithm to solve all of the above problems. The structural characterization says, roughly speaking, that a bipartite graph has a Pfaffian orientation if and only if it can be obtained by piecing together (in a specified way) planar bipartite graphs and one sporadic nonplanar bipartite graph.Comment: 47 pages, published versio

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

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

Hole의 관점에서 그래프와 유향그래프의 구조에 관한 연구

학위논문(박사)--서울대학교 대학원 :사범대학 수학교육과,2019. 8. 김서령.이 논문에서는 유향그래프와 그래프의 홀의 관점에서 계통발생 그래프와 그래프의 삼각화에 대하여 연구한다. 길이 4 이상인 유도된 싸이클을 홀이라 하고 홀이 없는 그래프를 삼각화된 그래프라 한다. 구체적으로, 싸이클을 갖지 않는 유향그래프의 계통발생 그래프가 삼각화된 그래프인지 판정하고, 주어진 그래프를 삼각화하여 클릭수가 크게 차이 나지 않는 그래프를 만드는 방법을 찾고자 한다. 이 논문은 연구 내용에 따라 두 부분으로 나뉜다. 먼저 $(1, i)$ 유향그래프와 $(i, 1)$ 유향그래프의 계통발생 그래프를 완전하게 특징화하고, $(2, j)$ 유향그래프 $D$의 모든 유향변에서 방향을 제거한 그래프가 삼각화된 그래프이면, $D$의 계통발생 그래프 역시 삼각화된 그래프임을 보였다. 또한 적은 수의 삼각형을 갖는 연결된 그래프의 계통발생수를 계산한 정리를 확장하여 많은 수의 삼각형을 포함한 연결된 그래프의 계통발생수를 계산하였다. 다른 한 편 그래프 $G$의 비삼각화 지수 $i(G)$에 대하여 $\omega(G^*)-\omega(G) \le i(G)$를 만족하는 $G$의 삼각화된 그래프 $G^*$가 존재함을 보였다. 그리고 이를 도구로 이용하여 NC property를 만족하는 그래프가 Hadwiger 추측과 Erd\H{o}s-Faber-Lov\'{a}sz 추측을 만족함을 증명하고, 비삼각화 지수가 유계인 그래프들이 linearly $\chi$-bounded임을 증명하였다.This thesis aims at studying phylogeny graphs and graph completions in the aspect of holes of graphs or digraphs. A hole of a graph is an induced cycle of length at least four and a graph is chordal if it does not contain a hole. Specifically, we determine whether the phylogeny graphs of acyclic digraphs are chordal or not and find a way of chordalizing a graph without increasing the size of maximum clique not so much. In this vein, the thesis is divided into two parts. In the first part, we completely characterize phylogeny graphs of $(1, i)$ digraphs and $(i,1)$ digraphs, respectively, for a positive integer $i$. Then, we 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 $j$. In addition, we extend the existing theorems computing phylogeny numbers of connected graph with a small number of triangles to results computing phylogeny numbers of connected graphs with many triangles. In the second part, we present a minimal chordal supergraph $G^*$ of a graph $G$ satisfying the inequality $\omega(G^*) - \omega(G) \le i(G)$ for the non-chordality index $i(G)$ of $G$. Using the above chordal supergraph as a tool, we prove that the family of graphs satisfying the NC property satisfies the Hadwiger conjecture and the Erd\H{o}s-Faber-Lov\'{a}sz Conjecture, and the family of graphs with bounded non-chordality indices is linearly $\chi$-bounded.Contents Abstract i 1 Introduction 1 1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Phylogeny graphs . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Graph colorings and chordal completions . . . . . . . . 14 2 Phylogeny graphs 19 2.1 Chordal phylogeny graphs . . . . . . . . . . . . . . . . . . . . 19 2.1.1 (1,j) phylogeny graphs and (i,1) phylogeny graphs . . 20 2.1.2 (2,j) phylogeny graphs . . . . . . . . . . . . . . . . . . 28 2.2 The phylogeny number and the triangles and the diamonds of a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 A new minimal chordal completion 61 3.1 Graphs with the NC property . . . . . . . . . . . . . . . . . . 64 3.2 The Erd˝ os-Faber-Lovász Conjecture . . . . . . . . . . . . . . . 73 3.3 A minimal chordal completion of a graph . . . . . . . . . . . . 80 3.3.1 Non-chordality indices of graphs . . . . . . . . . . . . . 80 3.3.2 Making a local chordalization really local . . . . . . . . 89 3.4 New χ-bounded classes . . . . . . . . . . . . . . . . . . . . . . 97 Abstract (in Korean) 107Docto