Related papers: The geodesic cover problem for butterfly networks
In this paper, we study the complexity of the edge monitoring problem. A vertex $v$ monitors an edge $e$ if both extremities together with $v$ form a triangle in the graph. Given a graph $G=(V,E)$ and a weight function on edges $c$ where…
We study the complexity of finding the \emph{geodetic number} on subclasses of planar graphs and chordal graphs. A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every vertex of $G$ lies in a shortest path between some pair…
A geometric graph is a graph embedded in the plane with vertices at points and edges drawn as curves (which are usually straight line segments) between those points. The average transversal complexity of a geometric graph is the number of…
A graph on 5 vertices consisting of 2 copies of the cycle graph C3 sharing a common vertex is called the Butterfly graph (B). The smallest natural number s such that any two-colouring (say red and blue) of the edges of Kj*s has a copy of a…
The problem of finding an optimal vertex cover in a graph is a classic NP-complete problem, and is a special case of the hitting set question. On the other hand, the hitting set problem, when asked in the context of induced geometric…
We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for {\em graphs with semi-edges}. The notion of graph covering is a discretization of coverings between surfaces or topological…
An isometric path between two vertices in a graph G is a shortest path joining them. The isometric-path number of G, denoted by ip(G), is the minimum number of isometric paths required to cover all vertices of G. In this paper, we determine…
A {\it path covering} of a graph $G$ is a set of vertex disjoint paths of $G$ containing all the vertices of $G$. The {\it path covering number} of $G$, denoted by $P(G)$, is the minimum number of paths in a path covering of $G$. An {\sl…
A graph $G$ is $k$-edge geodetic graph if every edge of $G$ lies in at least one geodesic of length $k$. We studied some basic properties of $k$-edge geodetic graphs. We investigated the $k$ edge-geodeticity of complete bipartite graph…
We consider discretization of the 'geometric cover problem' in the plane: Given a set $P$ of $n$ points in the plane and a compact planar object $T_0$, find a minimum cardinality collection of planar translates of $T_0$ such that the union…
A graph $G$ is geodetic if between any two vertices there exists a unique shortest path. In 1962 Ore raised the challenge to characterize geodetic graphs, but despite many attempts, such characterization still seems well beyond reach. We…
An overlap representation is an assignment of sets to the vertices of a graph in such a way that two vertices are adjacent if and only if the sets assigned to them overlap. The overlap number of a graph is the minimum number of elements…
The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover…
The "edge polytope" of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k =2, 3, 5 we determine the maximum number of vertices of…
A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$ vertices in…
The widely studied edge modification problems ask how to minimally alter a graph to satisfy certain structural properties. In this paper, we introduce and study a new edge modification problem centered around transforming a given graph into…
A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u,v) is at least…
A geometric graph is a graph whose vertices are points in general position in the plane and its edges are straight line segments joining these points. In this paper we give an $O(n^2 \log n)$ algorithm to compute the number of pairs of…
We analyze the computational complexity of the following computational problems called Bounded-Density Edge Deletion and Bounded-Density Vertex Deletion: Given a graph $G$, a budget $k$ and a target density $\tau_\rho$, are there $k$ edges…
The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex, connected graph is at least (1+o(1)) n…