Related papers: Computing graph gonality is hard

Given a graph $G$, an {\em obstacle representation} of $G$ is a set of points in the plane representing the vertices of $G$, together with a set of connected obstacles such that two vertices of $G$ are joined by an edge if and only if the…

The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $\mathscr D$ is $1+\max \mathscr D$.…

We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is its average…

We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. We show that equality holds for grid graphs and complete multipartite graphs. We prove that the treewidth lower bound also holds for…

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…

Local Irregularity Conjecture states that every simple connected graph, except special cacti, can be decomposed into at most three locally irregular graphs, i.e., graphs in which adjacent vertices have different degrees. The connected…

A derangement $k$-representation of a graph $G$ is a map $\pi$ of $V(G)$ to the symmetric group $S_k$, such that for any two vertices $v$ and $u$ of $V(G)$, $v $ and $u$ are adjacent if and only if $\pi(v)(i) \neq \pi(u)(i)$ for each $i \in…

The thickness $\theta(G)$ of a graph $G$ is the minimum number of planar spanning subgraphs into which the graph $G$ can be decomposed. As a topological invariant of a graph, it is a measurement of the closeness to planarity of a graph, and…

In this paper we investigate the extremal relationship between two well-studied graph parameters: the order of the largest homogeneous set in a graph $G$ and the maximal number of distinct degrees appearing in an induced subgraph of $G$,…

The Weisfeiler-Leman dimension of a graph $G$ is the least number $k$ such that the $k$-dimensional Weisfeiler-Leman algorithm distinguishes $G$ from every other non-isomorphic graph. The dimension is a standard measure of the descriptive…

We introduce the partition function of edge-colored graph homomorphisms, of which the usual partition function of graph homomorphisms is a specialization, and present an efficient algorithm to approximate it in a certain domain. Corollaries…

A graph $ G $ is minimally $ t $-tough if the toughness of $ G $ is $ t $ and deletion of any edge from $ G $ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $ t $-tough graph is $ \lceil 2t\rceil…

The connectivity of a graph is an important parameter to evaluate its reliability. $k$-restricted connectivity (resp. $R^h$-restricted connectivity) of a graph $G$ is the minimum cardinality of a set $S$ of vertices in $G$, if exists, whose…

A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges. The $t$-tessellability problem aims to decide whether there is a…

A vertex set $S$ of a graph $G$ is geodetic if every vertex of $G$ lies on a shortest path between two vertices in $S$. Given a graph $G$ and $k \in \mathbb N$, the NP-hard Geodetic Set problem asks whether there is a geodetic set of size…

Given a graph $G$ and a non trivial partition $(V_1,V_2)$ of its vertex-set, the satisfaction of a vertex $v\in V_i$ is the ratio between the size of it's closed neighborhood in $V_i$ and the size of its closed neighborhood in $G$. The…

For a graph invariant $\pi$, the Contraction($\pi$) problem consists in, given a graph $G$ and two positive integers $k,d$, deciding whether one can contract at most $k$ edges of $G$ to obtain a graph in which $\pi$ has dropped by at least…

Although NP-Complete problems are the most difficult decisional problems, it is possible to discover in them polynomial (or easy) observables. We study the Graph Partitioning Problem showing that it is possible to recognize in it two…

An edge-weighted graph $G=(V,E)$ is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as…

Planar graphs can be represented as intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin \& Gon{\c{c}}alves, 2009), \textsc{L}-shapes (Gon{\c{c}}alves et al, 2018).…