Related papers: Hull and Geodetic Numbers for Some Classes of Orie…

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 finite, simple, and undirected graph and let $S$ be a set of vertices of $G$. In the geodetic convexity, a set of vertices $S$ of a graph $G$ is convex if all vertices belonging to any shortest path between two vertices of $S$…

We prove three results conjectured or stated by Chartrand, Fink and Zhang [European J. Combin {\bf 21} (2000) 181--189, Disc. Appl. Math. {\bf 116} (2002) 115--126, and pre-print of ``The hull number of an oriented graph'']. For a digraph…

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…

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…

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 {\em Kneser graph} $K(2n+k,n)$, for positive integers $n$ and $k$, is the graph $G=(V,E)$ such that $V=\{S\subseteq\{1,\ldots,2n+k\} : |S|=n\}$ and there is an edge $uv\in E$ whenever $u\cap v=\emptyset$. Kneser graphs have a nice…

An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is…

In this work, for a given oriented graph $D$, we study its interval and hull numbers, respectively, in the oriented geodetic, P3 and P3* convexities. This last one, we believe to be formally defined and first studied in this paper, although…

A set of vertices $S$ of a graph $G$ is a geodetic set of $G$ if every vertex $v\not\in S$ lies on a shortest path between two vertices of $S$. The minimum cardinality of a geodetic set of $G$ is the geodetic number of $G$ and it is denoted…

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…

Let $G$ be a simple graph and let $S$ be a subset of its vertices. We say that $S$ is $P_3$-convex if every vertex $v \in V(G)$ that has at least two neighbors in $S$ also belongs to $S$. The $P_3$-hull set of $S$ is the smallest…

A maximal geodesic in a graph is a geodesic (alias shortest path) which is not a subpath of a longer geodesic. The geodesic-transversal problem in a graph $G$ is introduced as the task to find a smallest set $S$ of vertices of $G$ such that…

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 $K\subseteq{\mathbb R}^n$ be a convex semialgebraic set. The semidefinite extension degree ${\mathrm{sxdeg}}(K)$ of $K$ is the smallest number $d$ such that $K$ is a linear image of an intersection of finitely many spectrahedra, each of…

The stress interval $S(u,v)$ between $u,v\in V(G)$ is the set of all vertices in a graph $G$ that lie on every shortest $u,v$-path. A set $U \subseteq V(G)$ is stress convex if $S(u,v) \subseteq U$ for any $u,v\in U$. A vertex $v \in V(G)$…

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$…

A set $D$ of vertices in a graph $G$ is a dominating set if every vertex of $G$, which is not in $D$, has a neighbor in $D$. A set of vertices $D$ in $G$ is convex (respectively, isometric), if all vertices in all shortest paths…

A set $S$ of vertices of a graph $G$ is \emph{monophonic convex} if $S$ contains all the vertices belonging to any induced path connecting two vertices of $S$. The cardinality of a maximum proper monophonic convex set of $G$ is called the…

A path $P = v_1, ..., v_t$ is a {\em triangle path} (respectively, {\em monophonic path}) of $G$ if no edges exist joining vertices $v_i$ and $v_j$ of $P$ such that $|j - i| > 2$; (respectively, $|j - i| > 1$). A set of vertices $S$ is {\em…