Related papers: Exact-$2$-Relation Graphs
Let $G$ be a finite group. For a fixed element $g$ in $G$ and a given subgroup $H$ of $G$, the relative $g$-noncommuting graph of $G$ is a simple undirected graph whose vertex set is $G$ and two vertices $x$ and $y$ are adjacent if $x \in…
Let $G$ be a simple connected graph. If every pendant path in $G$ is at least $P_s$, we denote that $G\in \mathbb{G}_s$. For $G \in \mathbb{G}_s$, let $Q_s(G)$ be the set of vertices in $G$ that are distance $s$ from the pendant vertex, and…
An independent edge set of graph $G$ is a matching, and is maximal if it is not a proper subset of any other matching of $G$. The number of all the maximal matchings of $G$ is denoted by $\Psi(G)$. In this paper, an algorithm to count…
Many algorithms formulate graph matching as an optimization of an objective function of pairwise quantification of nodes and edges of two graphs to be matched. Pairwise measurements usually consider local attributes but disregard contextual…
Deciding whether a collection of unrooted trees is compatible is a fundamental problem in phylogenetics. Two different graph-theoretic characterizations of tree compatibility have recently been proposed. In one of these, tree compatibility…
A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…
A set $S$ of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in $S$ are disjoint. In this paper, we consider the graphs that have a unique maximum open packing. We characterize the trees with this…
A communication network can be modeled as a directed connected graph with edge weights that characterize performance metrics such as loss and delay. Network tomography aims to infer these edge weights from their pathwise versions measured…
Amalgamation in the totally non-negative part of positroid varieties is equivalent to gluing copies of $Gr^{TP}(1,3)$ and $Gr^{TP}(2,3)$. Lam has proposed to represent amalgamation in positroid varieties by equivalence classes of relations…
The vertex-random graphs called proximity catch digraphs (PCDs) have been introduced recently and have applications in pattern recognition and spatial pattern analysis. A PCD is a random directed graph (i.e., digraph) which is constructed…
An edge cut C of a graph G is tight if |C \M| = 1 for every perfect matching M of G. Barrier-cuts and 2-separation cuts, also referred to as ELP-cuts, are two important types of tight cuts in matching covered graphs. Edmonds, Lovasz and…
A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…
A connected graph can be associated with two distinct evolution algebras. In the first case, the structural matrix is the adjacency matrix of the graph itself. In the second case, the structural matrix is the transition probabilities matrix…
Evolutionary scenarios describing the evolution of a family of genes within a collection of species comprise the mapping of the vertices of a gene tree $T$ to vertices and edges of a species tree $S$. The relative timing of the last common…
A path $P$ in an edge-colored graph $G$ is called a proper path if no two adjacent edges of $P$ are colored the same, and $G$ is proper connected if every two vertices of $G$ are connected by a proper path in $G$. The proper connection…
A path $P$ in an edge-colored graph $G$ is a \emph{proper path} if no two adjacent edges of $P$ are colored with the same color. The graph $G$ is \emph{proper connected} if, between every pair of vertices, there exists a proper path in $G$.…
Let ${\cal G}=(G,w)$ be a weighted simple finite connected graph, that is, let $G$ be a simple finite connected graph endowed with a function $w$ from the set of the edges of $G$ to the set of real numbers. For any subgraph $G'$ of $G$, we…
In this paper we study graphs which admit acyclic orientations that contain a pair of arc-disjoint out-branching and in-branching (such an orientation is called good) and we focus on edge-minimal such graphs. A 2T-graph is a graph whose…
We consider three probability measures on subsets of edges of a given finite graph $G$, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the…
Leaf powers and pairwise compatibility graphs were introduced over twenty years ago as simplified graph models for phylogenetic trees. Despite significant research, several properties of these graph classes remain poorly understood. In this…