Related papers: Mutual-visibility problems in Kneser and Johnson g…

We study unit disk visibility graphs, where the visibility relation between a pair of geometric entities is defined by not only obstacles, but also the distance between them. That is, two entities are not mutually visible if they are too…

The vertex set of the Kneser graph $K(n,k)$ is $V = \binom{[n]}{k}$ and two vertices are adjacent if the corresponding sets are disjoint. For any graph $F$, the largest size of a vertex set $U \subseteq V$ such that $K(n,k)[U]$ is $F$-free,…

The variety of mutual-visibility problems contains four members, as does the variety of general position problems. The basic problem is to determine the cardinality of the largest such sets. In this paper, these eight invariants are…

In multiagent systems, effective coordination, coverage, and communication often rely on the concept of visibility between agents or nodes within the system. Graph-theoretically, for any subset $X$ of vertices of a graph $G$, two vertices…

The general position problem for graphs asks for the largest number of vertices in a subset $S \subseteq V(G)$ of a graph $G$ such that for any $u,v \in S$ and any shortest $u,v$-path $P$ we have $S \cap V(P) = \{ u,v\} $, whereas the…

The K-way vertex cut problem} consists in, given a graph G, finding a subset of vertices of a given size, whose removal partitions G into the maximum number of connected components. This problem has many applications in several areas. It…

For graph $G$, $F$ and integer $n$, the generalized Tu\'an number $ex(n,G,F)$ denotes the maximum number of copies of $G$ that an $F$-free $n$-vertex graph can have. We study this parameter when both $G$ and $F$ are complete bipartite…

Let $\Gamma$ be a straight-line drawing of a graph and let $u$ and $v$ be two vertices of $\Gamma$. The Gabriel disk of $u,v$ is the disk having $u$ and $v$ as antipodal points. A pair $\langle \Gamma_0,\Gamma_1 \rangle$ of vertex-disjoint…

A vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected. A super vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected and there…

The concept of mutual-visibility (MV) has been extended in several directions. A vertex subset $S$ of a graph $G$ is a $k$-distance mutual-visibility ($k$DMV) set if for any two vertices in $S$, there is a geodesic between them of length at…

For fixed graphs $F$ and $H$, the generalized Tur\'an problem asks for the maximum number $ex(n,H,F)$ of copies of $H$ that an $n$-vertex $F$-free graph can have. In this paper, we focus on cases with $F$ being $B_{r,s}$, the graph…

A graph $G=(V,E)$ is a {\it unipolar graph} if there exits a partition $V=V_1 \cup V_2$ such that, $V_1$ is a clique and $V_2$ induces the disjoint union of cliques. The complement-closed class of {\it generalized split graphs} are those…

A set of vertices $W$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W$. A metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. A bipartite graph G(n,n) is…

Visibility graph of a polygon corresponds to its internal diagonals and boundary edges. For each vertex on the boundary of the polygon, we have a vertex in this graph and if two vertices of the polygon see each other there is an edge…

Two vertices $u, v \in V$ of an undirected connected graph $G=(V,E)$ are resolved by a vertex $w$ if the distance between $u$ and $w$ and the distance between $v$ and $w$ are different. A set $R \subseteq V$ of vertices is a $k$-resolving…

We study the problem of finding a minimum $k$-critical-bipartite graph of order $(n,m)$: a bipartite graph $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical-bipartite, and the tuple $(|E|, \Delta_U, \Delta_V)$, where…

We consider the problem of determining if a pair of undirected graphs $\langle G_\mathsf{V}, G_\mathsf{H} \rangle$, which share the same vertex set, has a representation using opaque geometric shapes for vertices, and vertical/horizontal…

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric…

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…

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. To be more formal, a graph $G$ is a median graph if, for all $\mu, u,v\in…