Related papers: The path sequence of a graph

A graph G is a 2-tree if G=K_3, or G has a vertex v of degree 2, whose neighbours are adjacent, and G\v{i}s a 2-tree. A characterization of the degree sequences of 2-trees is given. This characterization yields a linear-time algorithm for…

Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…

The status of a vertex $x$ in a graph is the sum of the distances between $x$ and all other vertices. Let $G$ be a connected graph. The status sequence of $G$ is the list of the statuses of all vertices arranged in nondecreasing order. $G$…

A spanning subgraph of a graph G is called a [0,2]-factor of G, if for . is a union of some disjoint cycles, paths and isolate vertices, that span the graph G. It is easy to get a [0,2]-factor of G and there would be many of [0,2]-factors…

We consider the size of the smallest set of vertices required to intersect every longest path in a chordal graph. Such sets are known as longest path transversals. We show that if $\omega(G)$ is the clique number of a chordal graph $G$,…

The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…

The Shortest Path Reconfiguration problem has as input a graph G (with unit edge lengths) with vertices s and t, and two shortest st-paths P and Q. The question is whether there exists a sequence of shortest st-paths that starts with P and…

The dominating graph of a graph $H$ has as its vertices all dominating sets of $H$, with an edge between two dominating sets if one can be obtained from the other by the addition or deletion of a single vertex of $H$. In this paper we prove…

The crossing number of a graph $G$ is the least number of crossings over all possible drawings of $G$. We present a structural characterization of graphs with crossing number one.

For a countable ultrahomogeneous graph G let P(G) denote the collection of domains of subgraphs of G isomorphic to G. The order types of maximal chains in the set P(G) U {\o} ordered by the inclusion are characterized as: (I) the order…

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…

Directed acyclic graphs whose nodes are all the divisors of a positive integer $n$ and arcs $(a,b)$ defined by $a$ divides $b$ are considered. Fourteen graph invariants such as order, size, and the number of paths are investigated for two…

This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…

In 2006, Bar\'at and Thomassen posed the following conjecture: for each tree $T$, there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(G)|$ is divisible by $|E(T)|$, then $G$ admits a decomposition…

An $\textit{isometric path}$ is a shortest path between two vertices. An $\textit{isometric path partition}$ (IPP) of a graph $G$ is a set $I$ of vertex-disjoint isometric paths in $G$ that partition the vertices of $G$. The…

The $t$-connected ideal of a graph $G$ is generated by all connected induced subgraphs of $G$ with $t$ vertices. When $t = 2$, this coincides with the usual edge ideal of the graph. Following the work of Faridi et al., we give a…

For a graph $G$ and $a,b\in V(G)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and $b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and $b$ in $G$. Two vertices…

A graph is \emph{hamiltonian-connected} if every pair of vertices can be connected by a hamiltonian path, and it is \emph{hamiltonian} if it contains a hamiltonian cycle. We construct families of non-hamiltonian graphs for which the ratio…

A consistent path system in a graph $G$ is an collection of paths, with exactly one path between any two vertices in $G$. A path system is said to be consistent if it is intersection-closed. We say that $G$ is strictly metrizable if every…

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…