Related papers: Succinct Definitions in the First Order Theory of …

The boxicity of a graph $G=(V,E)$ is the smallest integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap \cdots \cap E_k$. In the first part of this note, we prove that every graph on $m$…

Let $G$ be a graph. A good function is a function $f:V(G)\rightarrow \{-1,1\}$, satisfying $f(N(v))\geq 1$, for each $v\in V(G)$, where $ N(v)=\{u\in V(G)\, |\, uv\in E(G) \} $ and $f(S) = \sum_{u\in S} f(u)$ for every $S \subseteq V(G) $.…

We propose new graph representations that exploit dense local structure to improve time and space simultaneously. Given an undirected graph $G$, we define a dual clique cover (DCC) representation of $G$ to be the pair $(C, L)$, where $C$ is…

Each relational structure X has an associated Gaifman graph, which endows X with the properties of a graph. Suppose that X is infinite, connected and of bounded degree. A first-order sentence in the language of X is almost surely true…

Let G be a simple graph without isolated vertices. For a vertex i in G, the degree d_i is the number of vertices adjacent to i and the average 2-degree m_i is the mean of the degrees of the vertices which are adjacent to i. The sequence of…

Alspach [ Bull. Inst. Combin. Appl., 52 (2008), pp. 7-20] defined the maximal matching sequencibility of a graph $G$, denoted $ms(G)$, to be the largest integer $s$ for which there is an ordering of the edges of $G$ such that every $s$…

We prove that graphs G, G' satisfy the same sentences of first-order logic with counting of quantifier rank at most k if and only if they are homomorphism-indistinguishable over the class of all graphs of tree depth at most k. Here G, G'…

We present a succinct data structure for permutation graphs, and their superclass of circular permutation graphs, i.e., data structures using optimal space up to lower order terms. Unlike concurrent work on circle graphs (Acan et al. 2022),…

Let G_<(n,p) denote the usual random graph G(n,p) on a totally ordered set of n vertices. We will fix p=1/2 for definiteness. Let L^< denote the first order language with predicates equality (x=y), adjacency (x~y) and less than (x<y). For…

A fair dominating set in a graph $G$ (or FD-set) is a dominating set $S$ such that all vertices not in $S$ are dominated by the same number of vertices from $S$; that is, every two vertices not in $S$ have the same number of neighbors in…

Let $D$ be an oriented graph. The inversion of a set $X$ of vertices in $D$ consists in reversing the direction of all arcs with both ends in $X$. The inversion number of $D$, denoted by ${\rm inv}(D)$, is the minimum number of inversions…

A subset of vertices in a graph $G$ is considered a maximal dissociation set if it induces a subgraph with vertex degree at most 1 and it is not contained within any other dissociation sets. In this paper, it is shown that for $n\geq 3$,…

Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of $F$ in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits,…

We show that for any natural number $s$, there is a constant $\gamma$ and a subgraph-closed class having, for any natural $n$, at most $\gamma^n$ graphs on $n$ vertices up to isomorphism, but no adjacency labeling scheme with labels of size…

We observe that a standard transformation between \emph{ordinal} trees (arbitrary rooted trees with ordered children) and binary trees leads to interesting succinct binary tree representations. There are four symmetric versions of these…

For a graph class $\mathcal{F}$, let $ex_{\mathcal{F}}(n)$ denote the maximum number of edges in a graph in $\mathcal{F}$ on $n$ vertices. We show that for every proper minor-closed graph class $\mathcal{F}$ the function…

Boxicity of a graph $G(V,$ $E)$, denoted by $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of axis parallel boxes in $\mathbb{R}^k$. The problem of computing boxicity is inapproximable even…

Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathscr{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$ appear together in an odd number of sets in $\mathscr{C}$. Let $c_2(G)$ denote the minimum…

For a graph $H$, let $c(H)=\inf\{c\,:\,e(G)\geq c|G| \mbox{ implies } G\succ H\,\}$, where $G\succ H$ means that $H$ is a minor of $G$. We show that if $H$ has average degree $d$, then $$ c(H)\le (0.319\ldots+o_d(1))|H|\sqrt{\log d} $$…

A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$…