Related papers: b-articulation points and b-bridges in strongly bi…

In this paper we introduce the edge inducibility problem. This is a common refinement of both the well known Kruskal--Katona theorem and the inducibility question introduced by Pippenger and Golumbic. Our first result is a hardness result.…

Let $G$ be a graph (with multiple edges allowed) and let $T$ be a tree in $G$. We say that $T$ is $\textit{even}$ if every leaf of $T$ belongs to the same part of the bipartition of $T$, and that $T$ is $\textit{weakly even}$ if every leaf…

A graph $G$ of order $nv$ where $n\geq 2$ and $v\geq 2$ is said to be weakly $(n,v)$-clique-partitioned if its vertex set can be decomposed in a unique way into $n$ vertex-disjoint $v$-cliques. It is strongly $(n,v)$-clique-partitioned if…

A class A of labelled graphs is bridge-addable if for all graphs G in A and all vertices u and v in distinct connected components of G, the graph obtained by adding an edge between u and u is also in A; the class A is monotone if for all G…

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by…

A labelling of a graph is an assignment of labels to its vertex or edge sets (or both), subject to certain conditions, a well established concept. A labelling of a graph G of order n is termed a numbering when the set of integers {1,...,n}…

We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S_3^{|E(G)|},…

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 Hawkes graph $\Gamma_H(G)$ of $G$ is the directed graph whose vertex set coincides with $\pi(G)$ and it has the edge $(p, q)$ whenever $q\in\pi(G/O_{p',p}(G))$. The Sylow graph $\Gamma_s(G)$ of $G$ is the directed graph with vertex set…

A Stick graph G=(A\cup B, E) is the intersection graph of a set A of horizontal segments and a set B of vertical segments in the plane, whose left and respectively bottom endpoints lie on the same ground line with slope -1. These endpoints…

An edge-coloured path is \emph{rainbow} if its edges have distinct colours. An edge-coloured connected graph is said to be \emph{rainbow connected} if any two vertices are connected by a rainbow path, and \emph{strongly rainbow connected}…

Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. Then the spectrum of $S(G^\sigma)$ is called the skew-spectrum of $G^\sigma$, denoted by $Sp_S(G^\sigma)$.…

For a graph $G$ with vertex set $V$, let N($G$) denote the number of nonempty subsets of $V$ that induce a connected graph in $G$. In this paper, we focus on determining N($G$) for $G$ in the family $\mathbb{B}_n$ of $n$-vertex bicyclic…

An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. The class of graphs which admit a VPT representation in a host tree with maximum degree at most h is denoted by [h,2,1]. The…

We call a pair of non-adjacent vertices in G a non-edge. Contraction of a non-edge {u, v} in G is the replacement of u and v with a single vertex z and then making all the vertices that are adjacent to u or v adjacent to z. A non-edge {u,…

In 1959, Erd\H{o}s and Gallai showed that every $2$-connected graph $G$ contains a cycle of length at least $\frac{2|E(G)|}{|V(G)|-1}$. This result was subsequently extended to weighted graphs by Bondy and Fan in 1991. A natural local…

Two previous papers, arXiv:1803.00284 and arXiv:1803.00281, introduced and studied strong subgraph $k$-connectivity of digraphs obtaining characterizations, lower and upper bounds and computational complexity results for the new digraph…

The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin…

Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax the conditions for graphs to be Hamiltonian by restricting Ore- and Fan-type degree…

A tuple (s1,t1,s2,t2) of vertices in a simple undirected graph is 2-linked when there are two vertex-disjoint paths respectively from s1 to t1 and s2 to t2. A graph is 2-linked when all such tuples are 2-linked. We give a new and simple…