Related papers: Minimum $2$-vertex-twinless connected spanning sub…

We consider a new problem of designing a network with small $s$-$t$ effective resistance. In this problem, we are given an undirected graph $G=(V,E)$, two designated vertices $s,t \in V$, and a budget $k$. The goal is to choose a subgraph…

Partitioning a connected graph into $k$~vertex-disjoint connected subgraphs of similar (or given) orders is a classical problem that has been intensively investigated since late seventies. Given a connected graph $G=(V,E)$ and a weight…

Subgraph counting aims to count occurrences of a template T in a given network G(V, E). It is a powerful graph analysis tool and has found real-world applications in diverse domains. Scaling subgraph counting problems is known to be memory…

Dense subgraph discovery is a fundamental problem in graph mining with a wide range of applications \cite{gionis2015dense}. Despite a large number of applications ranging from computational neuroscience to social network analysis, that take…

Given an edge-weighted directed graph $G=(V,E)$ on $n$ vertices and a set $T=\{t_1, t_2, \ldots, t_p\}$ of $p$ terminals, the objective of the \scss ($p$-SCSS) problem is to find an edge set $H\subseteq E$ of minimum weight such that $G[H]$…

We consider approximations for computing minimum weighted cuts in directed graphs. We consider both rooted and global minimum cuts, and both edge-cuts and vertex-cuts. For these problems we give randomized Monte Carlo algorithms that…

We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. It asks for a given graph to find a shortest path with minimum eccentricity. Although it is NP-hard in general graphs, we demonstrate that a…

The enumeration of minimal connected dominating sets is known to be notoriously hard for general graphs. Currently, it is only known that the sets can be enumerated slightly faster than $\mathcal{O}^{*}(2^n)$ and the algorithm is highly…

Given a graph of which the n vertices form a regular two-dimensional grid, and in which each (possibly weighted and/or directed) edge connects a vertex to one of its eight neighbours, the following can be done in O(scan(n)) I/Os, provided M…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. An edge subset $F\subseteq E(G)$ is called a restricted edge-cut if $G-F$ is disconnected and has no isolated vertices. The restricted edge-connectivity $\lambda'(G)$ of $G$ is…

The conditional $h$-vertex($h$-edge) connectivity of a connected graph $H$ of minimum degree $ k > h$ is the size of a smallest vertex(edge) set $F$ of $H$ such that $H - F$ is a disconnected graph of minimum degree at least $h.$ Let $G$ be…

The difference between the two largest eigenvalues of the adjacency matrix of a graph $G$ is called the spectral gap of $G.$ If $G$ is a regular graph, then its spectral gap is equal to algebraic connectivity. Abdi, Ghorbani and Imrich, in…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and $L(G)$ be the line graph of $G$, which has vertex set $E(G)$ and two vertices $e$ and $f$ of $L(G)$ is adjacent if $e$ and $f$ is incident in $G$. The vertex-edge graph…

We prove new results for approximating the graphic TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graphic TSP itself, we improve the approximation ratio to 7/5. For a…

Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson. Although the optimization and decision variants of finding connected vertex covers of minimum size or…

Let $G=(V,E)$ be a directed graph with $n$ vertices and $m$ edges. The graph $G$ is called singly-connected if for each pair of vertices $v,w \in V$ there is at most one simple path from $v$ to $w$ in $G$. Buchsbaum and Carlisle (1993) gave…

A vertex $v$ of a 2-connected cubic graph $G$ is $\lambda$-matchable if $G$ has a spanning subgraph in which $v$ has degree three whereas every other vertex has degree one, and we let $\lambda(G)$ denote the number of such vertices.…

The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and…

The geometric bottleneck Steiner network problem on a set of vertices $X$ embedded in a normed plane requires one to construct a graph $G$ spanning $X$ and a variable set of $k\geq 0$ additional points, such that the length of the longest…

Given an undirected weighted graph $G(V,E)$, a constrained sketch over a terminal set $T\subset V$ is a subgraph $G'$ that connects the terminal vertices while satisfying a given set of constraints. Examples include Steiner trees…