Related papers: Complexity of Metric Dimension on Planar Graphs

Let $\Gamma$ be a simple connected undirected graph with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$. The metric dimension of a graph $\Gamma$ is the least number of vertices in a set with the property that the list of distances from…

A set W \subseteq V (G) is called a resolving set, if for each pair of distinct vertices u,v \in V (G) there exists t \in W such that d(u,t) \neq d(v,t), where d(x,y) is the distance between vertices x and y. The cardinality of a minimum…

In this paper, we study planar drawings of maximal outerplanar graphs with the objective of achieving small height. A recent paper gave an algorithm for such drawings that is within a factor of 4 of the optimum height. In this paper, we…

Graph orientation is a well-studied area of graph theory. A proper orientation of a graph $G = (V,E)$ is an orientation $D$ of $E(G)$ such that for every two adjacent vertices $ v $ and $ u $, $ d^{-}_{D}(v) \neq d^{-}_{D}(u)$ where…

For a connected graph $G=(V,E)$, a matching $M\subseteq E$ is a matching cut of $G$ if $G-M$ is disconnected. It is known that for an integer $d$, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of…

Graph Burning asks, given a graph $G = (V,E)$ and an integer $k$, whether there exists $(b_{0},\dots,b_{k-1}) \in V^{k}$ such that every vertex in $G$ has distance at most $i$ from some $b_{i}$. This problem is known to be NP-complete even…

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted IDENTIFYING CODE, (OPEN) LOCATING-DOMINATING SET and METRIC DIMENSION) of an interval or a permutation graph. In…

The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In: Towards a…

The local metric dimension ${\rm dim}_l$ in relation to the clique number $\omega$ is investigated. It is proved that if $\omega(G)\leq n(G)-3$, then ${\rm dim}_l(G) \leq n(G)-3$ and the graphs attaining the bound classified. Moreover, the…

Algorithmic extension problems of partial graph representations such as planar graph drawings or geometric intersection representations are of growing interest in topological graph theory and graph drawing. In such an extension problem, we…

The metric dimension of a graph $G$ is the minimal size of a subset $R$ of vertices of $G$ that, upon reporting their graph distance from a distingished (source) vertex $v^\star$, enable unique identification of the source vertex $v^\star$…

The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…

In this paper, we investigate the approximability of two node deletion problems. Given a vertex weighted graph $G=(V,E)$ and a specified, or "distinguished" vertex $p \in V$, MDD(min) is the problem of finding a minimum weight vertex set $S…

We define a distance metric between partitions of a graph using machinery from optimal transport. Our metric is built from a linear assignment problem that matches partition components, with assignment cost proportional to transport…

We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the…

The modular product $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g',h')$ of $G\diamond H$ are adjacent if $g=g'$ and $hh'\in E(H)$, or $gg'\in E(G)$ and $h=h'$, or $gg'\in E(G)$…

We study the problem of computing the diameter and the mean distance of a continuous graph, i.e., a connected graph where all points along the edges, instead of only the vertices, must be taken into account. It is known that for continuous…

The dimension of a partially ordered set $P$ (poset for short) is the least positive integer $d$ such that $P$ is isomorphic to a subposet of $\mathbb{R}^d$ with the natural product order. Dimension is arguably the most widely studied…

We define, for any graph $G=(V,E)$, a boundary $\partial G \subseteq V$. The definition coincides with what one would expected for the discretization of (sufficiently nice) Euclidean domains and contains all vertices from the…

Let $G$ be a connected graph and $u,v$ and $w$ vertices of $G$. Then $w$ is said to {\em strongly resolve} $u$ and $v$, if there is either a shortest $u$-$w$ path that contains $v$ or a shortest $v$-$w$ path that contains $u$. A set $W$ of…