Related papers: Visibility in graphs under edge and vertex removal

For a graph $G$ and a parameter $k$, we call a vertex $k$-enabling if it belongs both to a clique of size $k$ and to an independent set of size $k$, and we call it $k$-excluding otherwise. Motivated by issues that arise in secret sharing…

Given a finite simple undirected graph $G$, let $T_1(G)$ denote the subset of vertices of $G$ such that every vertex of $T_1(G)$ belongs to at least one subgraph isomorphic to a graph obtained by connecting a single vertex to two vertices…

For a graph $G$, we define $\sigma_2(G)=min \{d(u)+d(v)| u,v\in V(G), uv\not\in E(G)\}$, or simply denoted by $\sigma_2$. A edge-colored graph is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct…

A subset $X$ of vertices in a graph $G$ is a {\em diameter 2 subset} if the distance of any two vertices of $X$ is at most two {\em in $G[X]$}. Relaxing this notion, a subset $X$ of vertices in a graph $G$ is a {\em 2-reachable subset} if…

Let $G=(V(G), E(G))$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. Let $S$ be a subset of $V(G)$, and let $B(S)$ be the set of neighbours of $S$ in $V(G) \setminus S$. The differential $\partial(S)$ of $S$ is the number…

An orientation of a graph $G$ is {\it in-out-proper} if any two adjacent vertices have different in-out-degrees, where the in-out-degree of each vertex is equal to the in-degree minus the out-degree of that vertex. The {\it in-out-proper…

Given a graph $G=(V,E)$, a set $S\subseteq V$ is said to be a monitoring edge-geodetic set if the deletion of any edge in the graph results in a change in the distance between at least one pair of vertices in $S$. The minimum size of such a…

We introduce a variation of metric dimension, called the multiset dimension. The representation multiset of a vertex $v$ with respect to $W$ (which is a subset of the vertex set of a graph $G$), $r_m (v|W)$, is defined as a multiset of…

Given graphs H_1,...,H_k, we study the minimum order of a graph G such that for each i, the induced copies of H_i in G cover V(G). We prove a general upper bound of twice the sum of the numbers m_i, where m_i is one less than the order of…

Let $G$ be a nontrivial connected and vertex-colored graph. A vertex subset $X$ is called rainbow if any two vertices in $X$ have distinct colors. The graph $G$ is called \emph{rainbow vertex-disconnected} if for any two vertices $x$ and…

Let $\beta>0$. Motivated by jumbled graphs defined by Thomason, the celebrated expander mixing lemma and Haemers's vertex separation inequality, we define that a graph $G$ with $n$ vertices is a weakly $(n,\beta)$-graph if $\frac{|X|…

A subset $F$ of edges in a connected graph $G$ is a $h$-extra edge-cut if $G-F$ is disconnected and every component has more than $h$ vertices. The $h$-extra edge-connectivity $\la^{(h)}(G)$ of $G$ is defined as the minimum cardinality over…

Let $G$ be a graph, and let $\lambda(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $\lambda(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on…

A subset $M$ of the edges of a graph $G$ is a matching if no two edges in $M$ are incident. A maximal matching is a matching that is not contained in a larger matching. A subset $S$ of vertices of a graph $G$ with no isolated vertices is a…

Let $G(V, E)$ be a finite, simple, isolate-free graph. A set $D$ of vertices of a graph $G$ with the vertex set $V$ is a double dominating set of $G$, if every vertex $v\in D$ has at least one neighbor in $D$ and every vertex $v \in V…

We consider undirected simple finite graphs. The sets of vertices and edges of a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. For a graph $G$, we denote by $\delta(G)$ and $\eta(G)$ the least degree of a vertex of $G$ and the…

The weak minor G of a graph G is the graph obtained from G by a sequence of edge-contraction operations on G. A weak-minor-closed family of upper embeddable graphs is a set G of upper embeddable graphs that for each graph G in G, every weak…

In an oriented graph, the inversion of a subset of vertices X is the operation reversing the direction of every arc with both endpoints in X. Given a graph G, the inversion distance between two orientations G is the minimum number of…

If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the…

Almost $4$-connectivity is a weakening of $4$-connectivity which allows for vertices of degree three. In this paper we prove the following theorem. Let $G$ be an almost $4$-connected triangle-free planar graph, and let $H$ be an almost…