Related papers: On the stress transit function

A tolled walk $T$ between two non-adjacent vertices $u$ and $v$ in a graph $G$ is a walk, in which $u$ is adjacent only to the second vertex of $T$ and $v$ is adjacent only to the second-to-last vertex of $T$. A toll interval between…

A walk $u_0u_1 \ldots u_{k-1}u_k$ of a graph $G$ is a \textit{weakly toll walk} if $u_0u_k \not\in E(G)$, $u_0u_i \in E(G)$ implies $u_i = u_1$, and $u_ju_k\in E(G)$ implies $u_j=u_{k-1}$. The {\em weakly toll interval} of a set $S…

Let $D$ be an orientation of a simple graph. Given $u,v\in V(D)$, a directed shortest $(u,v)$-path is a $(u,v)$-geodesic. $S \subseteq V(D)$ is convex if, for every $u,v \in S$, the vertices in each $(u,v)$-geodesic and in each…

A walk $W$ between vertices $u$ and $v$ of a graph $G$ is called a {\em tolled walk between $u$ and $v$} if $u$, as well as $v$, has exactly one neighbour in $W$. A set $S \subseteq V(G)$ is {\em toll convex} if the vertices contained in…

In this work, we introduce a new graph convexity, that we call Cycle Convexity, motivated by related notions in Knot Theory. For a graph $G=(V,E)$, define the interval function in the Cycle Convexity as $I_{cc}(S) = S\cup \{v\in V(G)\mid…

The stress of a vertex in a graph is the number of geodesics passing through it (A. Shimbel, 1953). A graph is $k$-stress regular if stress of each of its vertices is $k$. In this paper, we investigate some results and compute stress of…

Toll convexity is a variation of the so-called interval convexity. A tolled walk $T$ between $u$ and $v$ in $G$ is a walk of the form $T: u,w_1,\ldots,w_k,v,$ where $k\ge 1$, in which $w_1$ is the only neighbor of $u$ in $T$ and $w_k$ is…

A set S of vertices of a connected graph G is convex, if for any pair of vertices u; v 2 S, every shortest path joining u and v is contained in S . The convex hull CH(S) of a set of vertices S is defined as the smallest convex set in G…

Let $G$ be a graph and $S \subseteq V(G)$. In the cycle convexity, we say that $S$ is \textit{cycle convex} if for any $u\in V(G)\setminus S$, the induced subgraph of $S\cup\{u\}$ contains no cycle that includes $u$. The \textit{cycle…

The primary objective of this paper is to investigate the notions of geometric and sequential convexity within a graph-theoretic framework, with the aim of examining various structural properties and exploring the connection between these…

Let $G=(V_G,E_G)$ be a connected graph. The distance $d_G(u,v)$ between vertices $u$ and $v$ in $G$ is the length of a shortest $u-v$ path in $G$. The eccentricity of a vertex $v$ in $G$ is the integer $e_G(v)= \max\{ d_G(v,u) \colon u\in…

A walk $u_0u_1 \ldots u_{k-1}u_k$ is a \textit{weakly toll walk} if $u_0u_i \in E(G)$ implies $u_i = u_1$ and $u_ju_k\in E(G)$ implies $u_j=u_{k-1}$. A set $S$ of vertices of $G$ is {\it weakly toll convex} if for any two non-adjacent…

Given a graph $G = (V,E)$, a set $T$ of vertex pairs, and an integer $k$, Hitting Geodesic Intervals asks whether there is a set $S \subseteq V$ of size at most $k$ such that for each terminal pair $\{u,v\} \in T$, the set $S$ intersects at…

Let $D$ be a connected oriented graph. A set $S \subseteq V(D)$ is convex in $D$ if, for every pair of vertices $x, y \in S$, the vertex set of every $xy$-geodesic, ($xy$ shortest directed path) and every $yx$-geodesic in $D$ is contained…

Given an undirected graph $G=(V,E)$, a vertex $v\in V$ is edge-vertex (ev) dominated by an edge $e\in E$ if $v$ is either incident to $e$ or incident to an adjacent edge of $e$. A set $S^{ev}\subseteq E$ is an edge-vertex dominating set…

For a graph $G = (V(G), E(G))$, let $i(G)$ be the number of isolated vertices in $G$. The {\it isolated toughness} of $G$ is defined as $I(G) = min\{|S|/i(G-S) : S\subseteq V(G), i(G-S)\geq 2\}$ if $G$ is not complete; $I(G)=|V(G)|-1$…

The subject of graph convexity is well explored in the literature, the so-called interval convexities above all. In this work, we explore the cycle convexity, an interval convexity whose interval function is $I(S) = S \cup \{u \mid G[S \cup…

For a non-decreasing sequence of positive integers $S = (s_1,s_2,\ldots)$, the {\em $S$-packing chromatic number} $\chi_S(G)$ of $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $X_i$, $i \in…

Given a graph $G$, a subset $S \subseteq V(G)$ is \textit{cycle convex}, if for any vertex $v \in V(G) \setminus S$, the induced subgraph, $G[S \cup \{v\}]$ cannot form a cycle containing the vertex $v$. The \textit{exchange number} of $G$,…

In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph $G$ and an integer $r_{uv}$ for every pair of vertices $u,v\in V(G)$. The objective is to construct a subgraph $H$ of minimum weight which contains…