Interval graphs and interval orders

AbstractThis paper explores the intimate connection between finite interval graphs and interval orders. Special attention is given to the family of interval orders that agree with, or provide representations of, an interval graph. Two characterizations (one by P. Hanlon) of interval graphs with essentially unique agreeing interval orders are noted, and relationships between interval graphs and interval orders that concern the number of lengths required for interval representations and bounds on lengths of representing intervals are discussed.Two invariants of the family of interval orders that agree with an interval graph are established, namely magnitude, which affects end-point placements, and the property of having the lengths of all representing intervals between specified bounds. Extremization problems for interval graphs and interval orders are also considered

On powers of interval graphs and their orders

It was proved by Raychaudhuri in 1987 that if a graph power $G^{k-1}$ is an interval graph, then so is the next power $G^k$. This result was extended to $m$-trapezoid graphs by Flotow in 1995. We extend the statement for interval graphs by showing that any interval representation of $G^{k-1}$ can be extended to an interval representation of $G^k$ that induces the same left endpoint and right endpoint orders. The same holds for unit interval graphs. We also show that a similar fact does not hold for trapezoid graphs.Comment: 4 pages, 1 figure. It has come to our attention that Theorem 1, the main result of this note, follows from earlier results of [G. Agnarsson, P. Damaschke and M. M. Halldorsson. Powers of geometric intersection graphs and dispersion algorithms. Discrete Applied Mathematics 132(1-3):3-16, 2003]. This version is updated accordingl

The niche graphs of interval orders

The niche 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 and only if $N^+_D(x) \cap N^+_D(y) \neq \emptyset$ or $N^-_D(x) \cap N^-_D(y) \neq \emptyset$, where $N^+_D(x)$ (resp. $N^-_D(x)$) is the set of out-neighbors (resp. in-neighbors) of $x$ in $D$. A digraph $D=(V,A)$ is called a semiorder (or a unit interval order) if there exist a real-valued function $f:V \to \mathbb{R}$ on the set $V$ and a positive real number $\delta \in \mathbb{R}$ such that $(x,y) \in A$ if and only if $f(x) > f(y) + \delta$. A digraph $D=(V,A)$ is called an interval order if there exists an assignment $J$ of a closed real interval $J(x) \subset \mathbb{R}$ to each vertex $x \in V$ such that $(x,y) \in A$ if and only if $\min J(x) > \max J(y)$. S. -R. Kim and F. S. Roberts characterized the competition graphs of semiorders and interval orders in 2002, and Y. Sano characterized the competition-common enemy graphs of semiorders and interval orders in 2010. In this note, we give characterizations of the niche graphs of semiorders and interval orders.Comment: 7 page

Boxicity and Interval-Orders: Petersen and the Complements of Line Graphs

The boxicity of a graph is the smallest dimension $d$ allowing a representation of it as the intersection graph of a set of $d$-dimensional axis-parallel boxes. We present a simple general approach to determining the boxicity of a graph based on studying its ``interval-order subgraphs''. The power of the method is first tested on the boxicity of some popular graphs that have resisted previous attempts: the boxicity of the Petersen graph is $3$, and more generally, that of the Kneser-graphs $K(n,2)$ is $n-2$ if $n\ge 5$, confirming a conjecture of Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]. Since every line graph is an induced subgraph of the complement of $K(n,2)$, the developed tools show furthermore that line graphs have only a polynomial number of edge-maximal interval-order subgraphs. This opens the way to polynomial-time algorithms for problems that are in general $\mathcal{NP}$-hard: for the existence and optimization of interval-order subgraphs of line-graphs, or of interval-completions of their complement.Comment: 17 pages, 5 figures, appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

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