Related papers: Packing $T$-connectors in graphs needs more connec…
A stable set in a graph G is a set of mutually non-adjacent vertices, alpha(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T, of order…
A tree with at most k leaves is called k-ended tree, and a tree with exactly k leaves is called k-end tree, where a leaf is a vertex of degree one. Contraction of a graph G along the edge e means deleting the edge e and identifying its end…
We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is…
A bond in a graph is a minimal nonempty edge-cut. A connected graph $G$ is dual Hamiltonian if the vertex set can be partitioned into two subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are both trees. There is much…
In this series, we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects such as undirected graphs, directed graphs, bidirected graphs, hypergraphs and finitary matroids.…
We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gy\'arf\'as and Lehel from 1976…
We extend the edge version of the classical Menger's Theorem for undirected graphs to $n$-dimensional simplicial complexes with chains over the field $\mathbb{F}_2$. The classical Menger's Theorem states that two different vertices in an…
Tangles of graphs have been introduced by Robertson and Seymour in the context of their graph minor theory. Tangles may be viewed as describing "k-connected components" of a graph (though in a twisted way). They play an important role in…
Motivated by the question of how macromolecules assemble, the notion of an {\it assembly tree} of a graph is introduced. Given a graph $G$, the paper is concerned with enumerating the number of assembly trees of $G$, a problem that applies…
Given a set of terminals in 2D/3D, the network with the shortest total length that connects all terminals is a Steiner tree. On the other hand, with enough budget, every terminal can be connected to every other terminals via a straight…
We consider infinite connected quasi-transitive locally finite graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings'…
The generalized $k$-connectivity of a graph $G$, denoted by $\kappa_k(G)$, is the minimum number of internally edge disjoint $S$-trees for any $S\subseteq V(G)$ with $|S|=k$. The generalized $k$-connectivity is a natural extension of the…
We prove tight network topology dependent bounds on the round complexity of computing well studied $k$-party functions such as set disjointness and element distinctness. Unlike the usual case in the CONGEST model in distributed computing,…
In this short note we consider a variation of the connectivity Waiter-Client game $WC(n,q,\mathcal{A})$ played on an $n$-vertex graph $G$ which consists of $q+1$ disjoint spanning trees. In this game in each round Waiter offers Client $q+1$…
The search of spanning trees with interesting disjunction properties has led to the introduction of edge-disjoint spanning trees, independent spanning trees and more recently completely independent spanning trees. We group together these…
Designing well-connected graphs is a fundamental problem that frequently arises in various contexts across science and engineering. The weighted number of spanning trees, as a connectivity measure, emerges in numerous problems and plays a…
For a graph $G$, the mean subtree order of $G$ is the average order of a subtree of $G$. In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph $G$ and every…
We give a construction that provides infinitely many 2-connected, cubic, bipartite, and planar graphs G with 3k vertices and such that the number of disjoint copies of a 3-vertex path in G is less than k.
The chain theorem of Tutte states that every 3-connected graph can be constructed from a wheel $W_n$ by repeatedly adding edges and splitting vertices. It is not difficult to prove the following strengthening of this theorem: every…
For a connected graph $G$, the Wiener index, denoted by $W(G)$, is the sum of the distance of all pairs of distinct vertices and the eccentricity, denoted by $\varepsilon(G)$, is the sum of the eccentricity of individual vertices. In…