Related papers: Recognizing [h,2,1] graphs

For a connected graph G=(V,E), a subset U of V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. We show that the problem to test whether a graph has a disconnected cut is…

A labelling of a graph is an assignment of labels to its vertex or edge sets (or both), subject to certain conditions, a well established concept. A labelling of a graph G of order n is termed a numbering when the set of integers {1,...,n}…

Given a class of graphs $\mathcal{H}$, the problem $\oplus\mathsf{Sub}(\mathcal{H})$ is defined as follows. The input is a graph $H\in \mathcal{H}$ together with an arbitrary graph $G$. The problem is to compute, modulo $2$, the number of…

The Hom complex ${\rm Hom}(T,G)$ of graphs is a CW-complex associated to a pair of graphs $T$ and $G$, considered in the graph coloring problem. It is known that certain homotopy invariants of ${\rm Hom}(T,G)$ give lower bounds for the…

The $k$-Coloring problem on hereditary graph classes has been a deeply researched problem over the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs. We say that a…

We study the Induced $H$ Partition problem from the parameterized complexity point of view. In the Induced $H$ Partition problem the task is to partition vertices of a graph $G$ into sets $V_1,V_2,\dots,V_n$ such that the graph $H$ is…

The problem of packing as many subgraphs isomorphic to $H \in \mathcal H$ as possible in a graph for a class $\mathcal H$ of graphs is well studied in the literature. Both vertex-disjoint and edge-disjoint versions are known to be…

A dominating set of a graph is a set of vertices such that every vertex not in the set has at least one neighbor in the set. The problem of counting dominating sets is #P-complete for chordal graphs but solvable in polynomial time for its…

The spectrum of the $k$-power hypergraph of a graph $G$ is called the $k$-ordered spectrum of $G$.If graphs $G_1$ and $G_2$ have same $k$-ordered spectrum for all positive integer $k\geq2$, $G_1$ and $G_2$ are said to be high-ordered…

Recent results show that the structural similarity of graphs can be characterized by counting homomorphisms to them: the Tree Theorem states that the well-known color-refinement algorithm does not distinguish two graphs G and H if and only…

For a graph $G$, the tree graph ${\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of…

The Connected Vertex Cover problem is to decide if a graph G has a vertex cover of size at most $k$ that induces a connected subgraph of $G$. This is a well-studied problem, known to be NP-complete for restricted graph classes, and, in…

For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much…

Ordered matchings, defined as graphs with linearly ordered vertices, where each vertex is connected to exactly one edge, play a crucial role in the area of ordered graphs and their homomorphisms. Therefore, we consider related problems from…

Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C_H(G). Previous results cover only a few specific instances of this general…

For fixed integers $r,\ell \geq 0$, a graph $G$ is called an {\em $(r,\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. The class of $(r, \ell)$ graphs generalizes $r$-colourable graphs…

An \emph{s-graph} is a graph with two kinds of edges: \emph{subdivisible} edges and \emph{real} edges. A \emph{realisation} of an s-graph $B$ is any graph obtained by subdividing subdivisible edges of $B$ into paths of arbitrary length (at…

In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall…

Given nonnegative integers, $s$ and $k$, an $(s,k)$-polar partition of a graph $G$ is a partition $(A,B)$ of $V_G$ such that $G[A]$ and $\overline{G[B]}$ are complete multipartite graphs with at most $s$ and $k$ parts, respectively. If $s$…

As set systems, hypergraphs are omnipresent and have various representations ranging from Euler and Venn diagrams to contact representations. In a geometric representation of a hypergraph $H=(V,E)$, each vertex $v\in V$ is associated with a…