Related papers: Generalization of edge general position problem
The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the…
Graph theoretical problems based on shortest paths are at the core of research due to their theoretical importance and applicability. This paper deals with the geodetic number which is a global measure for simple connected graphs and it…
Covering problems are classical computational problems concerning whether a certain combinatorial structure 'covers' another. For example, the minimum vertex covering problem aims to find the smallest set of vertices in a graph so that each…
The \emph{general position problem} in graphs asks for a largest set of vertices in which no three lie on a common shortest path. The \emph{mutual-visibility problem} seeks a largest set of vertices such that every pair is connected by a…
A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${\rm…
We study the problem of deleting the smallest set $S$ of vertices (resp. edges) from a given graph $G$ such that the induced subgraph (resp. subgraph) $G \setminus S$ belongs to some class $\mathcal{H}$. We consider the case where graphs in…
A subset $R\subseteq V(G)$ of a graph $G$ is a general position set if any triple set $R_0$ of $R$ is non-geodesic in $G$, that is, no vertex of $R_0$ lies on any geodesic between the other two vertices of $R_0$ in $G$. Let $\mathcal{R}$ be…
A k-clique covering of a simple graph G, is an edge covering of G by its cliques such that each vertex is contained in at most k cliques. The smallest k for which G admits a k-clique covering is called local clique cover number of G and is…
For an integer $k\ge 3$, a $k$-path vertex cover of a graph $G=(V,E)$ is a set $T\subseteq V$ that shares a vertex with every path subgraph of order $k$ in $G$. The minimum cardinality of a $k$-path vertex cover is denoted by $\psi_k(G)$.…
Given an undirected graph G, the edge orientation problem asks for assigning a direction to each edge to convert G into a directed graph. The aim is to minimize the maximum out degree of a vertex in the resulting directed graph. This…
The minimum dominating set problem asks for a dominating set with minimum size. First, we determine some vertices contained in the minimum dominating set of a graph. By applying a particular scheme, we ensure that the resulting graph is…
We introduce a new graph-theoretic concept in the area of network monitoring. In this area, one wishes to monitor the vertices and/or the edges of a network (viewed as a graph) in order to detect and prevent failures. Inspired by two…
We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the…
A set of vertices $X$ of a graph $G$ is a strong edge geodetic set if to any pair of vertices from $X$ we can assign one (or zero) shortest path between them such that every edge of $G$ is contained in at least one on these paths. The…
A monitoring edge-geodetic set, or simply an MEG-set, of a graph $G$ is a vertex subset $M \subseteq V(G)$ such that given any edge $e$ of $G$, $e$ lies on every shortest $u$-$v$ path of $G$, for some $u,v \in M$. The monitoring…
We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We…
We consider a generalized poset sorting problem (GPS), in which we are given a query graph $G = (V, E)$ and an unknown poset $\mathcal{P}(V, \prec)$ that is defined on the same vertex set $V$, and the goal is to make as few queries as…
Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E' of E which dominates all edges of G. In particular, if every edge of G is dominated by…
We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed $\alpha$ between zero and one we are given a graph and two numbers $k \in \mathbb{N}$ and $t \in \mathbb{Q}$. The…
An edge dominating set of a graph G=(V,E) is a subset M of edges in the graph such that each edge in E-M is incident with at least one edge in M. In an instance of the parameterized edge dominating set problem we are given a graph G=(V,E)…