Related papers: Complexity results on locally-balanced $2$-partiti…

We study a family of graph modification problems called the F-Vertex Splitting problem. Given a graph G, the task is to determine whether G can be transformed into a graph G-prime belonging to a graph class F through a sequence of at most k…

The \emph{$r$-neighbourhood complexity} of a graph $G$ is the function counting, for a given integer $k$, the largest possible number, over all vertex-subsets $A$ of size $k$, of subsets of $A$ realized as the intersection between the…

An internal partition of an $n$-vertex graph $G=(V,E)$ is a partition of $V$ such that every vertex has at least as many neighbors in its own part as in the other part. It has been conjectured that every $d$-regular graph with $n>N(d)$…

Consider a graph $G = (V, E)$ and a function $f: V \rightarrow \{0, 1, 2\}$. A vertex $u$ with $f(u)=0$ is defined as \emph{undefended} by $f$ if it lacks adjacency to any vertex with a positive $f$-value. The function $f$ is said to be a…

We study edge partitions of a bipartite graph into induced-$2K_2$-free bipartite graphs, i.e.\ into Ferrers (chain) graphs. We define $\fp(G)$ as the minimum number of parts in such a partition. We prove general lower and upper bounds in…

Given a graph $G$ and a parameter $k$, the $k$-biclique problem asks whether $G$ contains a complete bipartite subgraph $K_{k,k}$. This is the most easily stated problem on graphs whose parameterized complexity is still unknown. We provide…

Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect…

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number…

Given an undirected graph $G$ and $q$ integers $n_1,n_2,n_3, \cdots, n_q$, balanced connected $q$-partition problem ($BCP_q$) asks whether there exists a partition of the vertex set $V$ of $G$ into $q$ parts $V_1,V_2,V_3,\cdots, V_q$ such…

Let $\mathcal{D}$ be a set of straight-line segments in the plane, potentially crossing, and let $c$ be a positive integer. We denote by $P$ the union of the endpoints of the straight-line segments of $\mathcal{D}$ and of the intersection…

For a graph $F$, a graph $G$ is \emph{$F$-free} if it does not contain an induced subgraph isomorphic to $F$. For two graphs $G$ and $H$, an \emph{$H$-coloring} of $G$ is a mapping $f:V(G)\rightarrow V(H)$ such that for every edge $uv\in…

Let $G(V,E)$ be a simple, undirected and connected graph. A dominating set $S \subseteq V(G)$ is called a $2$-\textit{secure dominating set} ($2$-SDS) in $G$, if for every pair of distinct vertices $u_1,u_2 \in V(G)$ there exists a pair of…

A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of ${\rm Aut}(G)$. Regular…

A graph covering projection, also referred to as a locally bijective homomorphism, is a mapping between the vertices and edges of two graphs that preserves incidences and is a local bijection. This concept originates in topological graph…

A graph $G = (V,E)$ is $\textit{monopolar}$ if its vertex set admits a partition $V = (C \uplus{} I)$ where $G[C]$ is a $\textit{cluster graph}$ and $I$ is an $\textit{independent set}$ in $G$; this is a \textit{monopolar partition} of $G$.…

A graph $\Gamma$ is called locally finite if, for each vertex $v$ of $\Gamma$, the set $\Gamma(v)$ of all neighbors of $v$ in $\Gamma$ is finite. For any locally finite graph $\Gamma$ with vertex set $V(\Gamma)$ and for any field $F$, let…

Neighborhood-prime labeling is a variation of prime labeling. A labeling $f:V(G) \to [|V(G)|]$ is a neighborhood-prime labeling if for each vertex $v\in V(G)$ with degree greater than $1$, the greatest common divisor of the set of labels in…

Let $G=(V,E)$ be a simple undirected graph. The open neighbourhood of a vertex $v$ in $G$ is defined as $N_G(v)=\{u\in V~|~ uv\in E\}$; whereas the closed neighbourhood is defined as $N_G[v]= N_G(v)\cup \{v\}$. For an integer $k$, a subset…

For a graph $G=(V,E)$, a set $D \subseteq V$ is called a semitotal dominating set of $G$ if $D$ is a dominating set of $G$, and every vertex in $D$ is within distance~$2$ of another vertex of~$D$. The \textsc{Minimum Semitotal Domination}…

Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not…