Related papers: $2$-blocks in strongly biconnected directed graphs

A graph $G$ on $n$ vertices is Hamiltonian if it contains a spanning cycle, and pancyclic if it contains cycles of all lengths from 3 to $n$. In 1984, Fan presented a degree condition involving every pair of vertices at distance two for a…

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…

A {\it semi-proper orientation} of a given graph $G$ is a function $(D,w)$ that assigns an orientation $D(e)$ and a positive integer weight $ w(e)$ to each edge $e$ such that for every two adjacent vertices $v$ and $u$, $S_{(D,w)}(v) \neq…

A Graph is called 2-self-centered if its diameter and radius both equal to 2. In this paper, we begin characterizing these graphs by characterizing edge-maximal 2-self-centered graphs via their complements. Then we split characterizing…

Let $G$ be a graph, and let $w: V(G) \to \mathbb{R}$ be a weight function on the vertices of $G$. For every subset $X$ of $V(G)$, let $w(X)=\sum_{v \in X} w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, for…

Let $G$ be a graph and $A$ be its adjacency matrix. A graph $G$ is invertible if its adjacency matrix $A$ is invertible and the inverse of $G$ is a weighted graph with adjacency matrix $A^{-1}$. A signed graph $(G,\sigma)$ is a weighted…

A 2-edge-coloured graph $G$ is {\bf supereulerian} if $G$ contains a spanning closed trail in which the edges alternate in colours. An {\bf eulerian factor} of a 2-edge-coloured graph is a collection of vertex disjoint induced subgraphs…

Suppose that $D=(V,E)$ is a strongly connected digraph. Let $u,v\in V(D)$. The maximum distance $md (u,v)$ is defined as $md(u,v)$=max\{$\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)$\} where $\overrightarrow{d}(u,v)$ denote the length…

A $d$-dimensional framework is a pair $(G,p)$, where $G=(V,E)$ is a graph and $p$ is a map from $V$ to $\mathbb{R}^d$. The length of an edge $xy\in E$ in $(G,p)$ is the distance between $p(x)$ and $p(y)$. A vertex pair $\{u,v\}$ of $G$ is…

Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent, alpha(G) is the size of a maximum independent set, and core(G) is the intersection of all maximum independent sets. The number d(X)=|X|-|N(X)| is the…

Given a connected graph $G$, a vertex $w\in V(G)$ strongly resolves two vertices $u,v\in V(G)$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $S$ of vertices is a strong metric…

The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$ is a quantitative measure of its $d$-dimensional rigidity, defined in terms of the eigenvalues of stiffness matrices associated with different embeddings of the graph…

Let $G_S$ be a graph with loops obtained from a graph $G$ of order $n$ and loops at $S \subseteq V(G)$. In this paper, we establish a neccesary and sufficient condition on the bipartititeness of a connected graph $G$ and the spectrum…

A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices,…

A vertex $w$ of a connected graph $G$ strongly resolves two vertices $u,v\in V(G)$, if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $S$ of vertices is a strong metric generator for…

A connected graph $G$ with at least $2m + 2n + 2$ vertices which contains a perfect matching is $E(m, n)$-{\it extendable}, if for any two sets of disjoint independent edges $M$ and $N$ with $|M| = m$ and $|N|= n$, there is a perfect…

Let $G=(V,E)$ be a simple connected graph. A connected edge cover of $G$ is a subset $S\subseteq E$ such that every vertex of $G$ is incident with at least one edge in $S$ and the subgraph induced by $S$ is connected. The connected edge…

Let $G=(V,E)$ be a finite undirected graph. If $P$ is an oriented path from $r_1\in V$ to $r_2\in V$, we define $\partial(P) = r_2-r_1$. If $R, S\subseteq V$, we denote by $P(G; R, S)$ the span of the set of all $\partial P\otimes \partial…

The toughness of a graph $G$ is defined as the largest real number $t$ such that for any set $S\subseteq V(G)$ such that $G-S$ is disconnected, $S$ has at least $t$ times more elements than $G-S$ has components (unless $G$ is complete, in…

We consider problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs. For the former problem, we show the minimum number of additional signs and give a linear-time…