Related papers: Tri-connectivity Augmentation in Trees

For integer $k\geq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|\geq k+1$ and given any subset $S\subseteq V(G)$ with $|S|=k,$ $G$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T.$ Thus a graph is…

Given a simple connected graph $G = (V, E)$, we seek to partition the vertex set $V$ into $k$ non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximum…

A vertex of degree one is called an end-vertex, and an end-vertex of a tree is called a leaf. A tree with at most $k$ leaves is called a $k$-ended tree. For a positive integer $k$, let $t_k$ be the order of a largest $k$-ended tree. Let…

Boesch and Chen (SIAM J. Appl. Math., 1978) introduced the cut-version of the generalized edge-connectivity, named $k$-edge-connectivity. For any integer $k$ with $2\leq k\leq n$, the {\em $k$-edge-connectivity} of a graph $G$, denoted by…

Given a $2$-vertex-twinless connected directed graph $G=(V,E)$, the minimum $2$-vertex-twinless connected spanning subgraph problem is to find a minimum cardinality edge subset $E^{t} \subseteq E$ such that the subgraph $(V,E^{t})$ is…

Let $G=(V,E)$ be a strongly connected graph with $|V|\geq 3$. For $T\subseteq V$, the strongly connected graph $G$ is $2$-T-connected if $G$ is $2$-edge-connected and for each vertex $w$ in $T$, $w$ is not a strong articulation point. This…

A vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected. A super vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected and there…

Constructing the maximum spanning tree $T$ of an edge-weighted connected graph $G$ is one of the important research topics in computer science and optimization, and the related research results have played an active role in practical…

The complexity of the maximum common connected subgraph problem in partial $k$-trees is still not fully understood. Polynomial-time solutions are known for degree-bounded outerplanar graphs, a subclass of the partial $2$-trees. On the other…

Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color appear with fewer than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper…

For any graph $G$, let $t(G)$ be the number of spanning trees of $G$, $L(G)$ be the line graph of $G$ and for any non-negative integer $r$, $S_r(G)$ be the graph obtained from $G$ by replacing each edge $e$ by a path of length $r+1$…

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 $l > 1$, the $l$-edge-connectivity $\kappa'_l(G)$ of a connected graph $G$ is defined as the minimum number of edges whose removal leaves a graph with at least $l$ components. A graph is minimally $(k,l)$-edge-connected if…

In 2010, Mader [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.] proved that every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor+m-1$ contains a path $P$ of…

A connected graph is 2K2-free if it does not contain a pair of independent edges as an induced subgraph. In this paper, we present the structural characterization of minimal vertex separator and show that there are polynomial number of…

We introduce three new cut tree structures of graphs $G$ in which the vertex set of the tree is a partition of $V(G)$ and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently,…

Fiber graphs of Gr\"obner bases from contingency tables are important in statistical hypothesis testing, where one studies random walks on these graphs using the Metropolis-Hastings algorithm. The connectivity of the graphs has implications…

Let $S\subseteq V(G)$ and $\pi_{G}(S)$ denote the maximum number $t$ of edge-disjoint paths $P_{1},P_{2},\ldots,P_{t}$ in a graph $G$ such that $V(P_{i})\cap V(P_{j})=S$ for any $i,j\in\{1,2,\ldots,t\}$ and $i\neq j$. If $S=V(G)$, then…

Many applications produce massive complex networks whose analysis would benefit from parallel processing. Parallel algorithms, in turn, often require a suitable network partition. For solving optimization tasks such as graph partitioning on…

Motivated by the increasing need to understand the algorithmic foundations of distributed large-scale graph computations, we study a number of fundamental graph problems in a message-passing model for distributed computing where $k \geq 2$…