Related papers: Navigating Between Packings of Graphic Sequences

The notion of graph covers (also referred to as locally bijective homomorphisms) plays an important role in topological graph theory and has found its computer science applications in models of local computation. For a fixed target graph…

A graph is subcubic if it is connected and its maximum vertex degree does not exceed 3. Two disjoint vertex subsets of a graph $G$ form a connected coalition in $G$ if neither of them is a connected dominating set but their union is a…

A sequence $S$ is potentially $K_{p_{1},p_{2},...,p_{t}}$ graphical if it has a realization containing a $K_{p_{1},p_{2},...,p_{t}}$ as a subgraph, where $K_{p_{1},p_{2},...,p_{t}}$ is a complete t-partite graph with partition sizes…

List packing is a notion that was introduced in 2021 (by Cambie et al.). The list packing number of a graph $G$, denoted $\chi_{\ell}^*(G)$, is the least $k$ such that for any list assignment $L$ that assigns $k$ colors to each vertex of…

Given a planar graph $G$, we consider drawings of $G$ in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding $\pi$ of the vertex set of $G$ into…

A collection of graphs is \textit{nearly disjoint} if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$, then the following holds. For every…

A subset $S$ of vertices of a graph $G=(V,E)$ is called a $k$-path vertex cover if every path on $k$ vertices in $G$ contains at least one vertex from $S$. Denote by $\psi_k(G)$ the minimum cardinality of a $k$-path vertex cover in $G$ and…

Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two…

For a degree sequence, we define the set of edges that appear in every labeled realization of that sequence as forced, while the edges that appear in none as forbidden. We examine structure of graphs whose degree sequences contain either…

In Partition Into Complementary Subgraphs (Comp-Sub) we are given a graph $G=(V,E)$, and an edge set property $\Pi$, and asked whether $G$ can be decomposed into two graphs, $H$ and its complement $\overline{H}$, for some graph $H$, in such…

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $\Pi_1,\ldots,\Pi_k$, where $\Pi_i$, $i\in [k]$, is an $i$-packing. The following…

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property $\Phi$. What happens if this question is modified in a way that we get a possibly infinite family of graphs…

We resolve in the affirmative conjectures of Repovs and A. Skopenkov (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our…

Suppose a finite, unweighted, combinatorial graph $G = (V,E)$ is the union of several (degree-)regular graphs which are then additionally connected with a few additional edges. $G$ will then have only a small number of vertices $v \in V$…

Let $P$ and $Q$ be finite point sets of the same cardinality in $\mathbb{R}^2$, each labelled from $1$ to $n$. Two noncrossing geometric graphs $G_P$ and $G_Q$ spanning $P$ and $Q$, respectively, are called compatible if for every face $f$…

We prove that deciding whether a given input word contains as subsequence every possible permutation of integers $\{1,2,\ldots,n\}$ is coNP-complete. The coNP-completeness holds even when given the guarantee that the input word contains as…

A permutation graph $G_\pi$ is a simple graph with vertices corresponding to the elements of $\pi$ and an edge between $i$ and $j$ when $i$ and $j$ are inverted in $\pi$. A set of vertices $D$ is said to dominate a graph $G$ when every…

For positive integers $\ell$ and $k$, a $(1^\ell, 2^k)$-packing edge-coloring of a graph $G$ is a partition of $E(G)$ into $\ell$ matchings and $k$ induced matchings. A graph is $d$-irregular if it has no adjacent vertices of degree $d$.…

Paths $P_1,\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of…

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$…