Related papers: The Restricted Edge-Connectivity of Strong Product…

A tree $T$ in an edge-colored graph is a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be a fixed integer with $2\le k\le n$. For a vertex subset $S \subseteq…

In this paper, we consider the question of computing sparse subgraphs for any input directed graph $G=(V,E)$ on $n$ vertices and $m$ edges, that preserves reachability and/or strong connectivity structures. We show $O(n+\min\{|{\cal…

A subgraph $H$ of a multigraph $G$ is called strongly spanning, if any vertex of $G$ is not isolated in $H$, while it is called maximum $k$-edge-colorable, if $H$ is proper $k$-edge-colorable and has the largest size. We introduce a…

Let $G$ be a contraction critically quasi $5$-connected graph on at least $14$ vertices. If there is a vertex $x\in V_{4}(G)$ such that $G[N_{G}(x)]\cong K_{1,3}$ or $G[N_{G}(x)]\cong C_{4}$, then $G$ has a quasi $5$-contractible subgraph…

For any pair of edges $e,f$ of a graph $G$, we say that {\em $e,f$ are $P_3$-connected in $G$} if there exists a sequence of edges $e=e_0,e_1,\ldots, e_k=f$ such that $e_i$ and $e_{i+1}$ are two edges of an induced $3$-vertex path in $G$…

Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G,r)$, of…

A graph $G$ is said to be $d$-distinguishable if there is a vertex coloring of $G$ with a set of $d$ colors which breaks all of the automorphisms of $G$ but the identity. We call the minimum $d$ for which a graph $G$ is $d$-distinguishiable…

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$.…

Due to the increasing discovery and implementation of networks within all disciplines of life, the study of subgraph connectivity has become increasingly important. Motivated by the idea of community (or sub-graph) detection within a…

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…

The leap eccentric connectivity index of $G$ is defined as $$L\xi^{C}(G)=\sum_{v\in V(G)}d_{2}(v|G)e(v|G)$$ where $d_{2}(v|G) $ be the second degree of the vertex $v$ and $e(v|G)$ be the eccentricity of the vertex $v$ in $G$. In this paper,…

This paper introduces the notion of an $(\iota,q)$-critical graph. The isolation number of a graph $G$, denoted by $\iota(G)$ and also known as the vertex-edge domination number of $G$, is the size of a smallest subset $D$ of the vertex set…

In this paper we study a new product of graphs called {\em tight product}. A graph $H$ is said to be a tight product of two (undirected multi) graphs $G_1$ and $G_2$, if $V(H)=V(G_1)\times V(G_2)$ and both projection maps $V(H)\to V(G_1)$…

Let $\mathbf{G}=\{G_1,\dots,G_{n-1}\}$ be a collection of not necessarily distinct $n$-vertex graphs with the same vertex set $V$. A path $P$ with $V(P)\subseteq V$ and $|E(P)|\leq n-1$ is rainbow in $\mathbf{G}$, if there exists an…

Over the past half century, the rigidity of graphs in $R^2$ has aroused a great deal of interest. Lov\'{a}sz and Yemini (1982) proved that every $6$-connected graph is rigid in $R^2$. Jackson and Jord\'{a}n (2005) provided a similar…

We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut…

A set of edges $X\subseteq E(G)$ of a graph $G$ is an edge general position set if no three edges from $X$ lie on a common shortest path. The edge general position number ${\rm gp}_{\rm e}(G)$ of $G$ is the cardinality of a largest edge…

The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. In this paper, the minimum degree of power graphs of certain classes of cyclic groups, abelian…

For distinct vertices $u$ and $v$ in a graph $G$, the {\em connectivity} between $u$ and $v$, denoted $\kappa_G(u,v)$, is the maximum number of internally disjoint $u$--$v$ paths in $G$. The {\em average connectivity} of $G$, denoted…

A graph $G$ is said to be $k$-$\gamma_{c}$-critical if the connected domination number $\gamma_{c}(G)$ is equal to $k$ and $\gamma_{c}(G + uv) < k$ for any pair of non-adjacent vertices $u$ and $v$ of $G$. Let $\zeta$ be the number of cut…