Related papers: On Making Directed Graphs Eulerian

Connectivity of temporal graphs has been widely studied both as graph theory and as gossip theory. In particular, it is well known that in order to connect every vertex to every other, a temporal graph needs to have at least $2n-4$ edges…

The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives a proof that, for graphs where each…

A monitoring edge-geodetic set of a graph is a subset $M$ of its vertices such that for every edge $e$ in the graph, deleting $e$ increases the distance between at least one pair of vertices in $M$. We study the following computational…

Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching}…

We aim to find orientations of mixed graphs optimizing the total reachability, a problem that has applications in causality and biology. For given a digraph $D$, we use $P(D)$ for the set of ordered pairs of distinct vertices in $V(D)$ and…

Let $G$ be a graph on $n$ vertices with adjacency matrix $A$, and let $\mathbf{1}$ be the all-ones vector. We call $G$ controllable if the set of vectors $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ spans the whole space…

Let $G=(V, E)$ be a graph where $V(G)$ and $E(G)$ are the vertex and edge sets, respectively. In a graph $G$, two edges $e_1, e_2\in E(G)$ are said to have a \emph{common edge} $e\neq e_1, e_2$ if $e$ joins an endpoint of $e_1$ to an…

We consider the \textsc{Steiner Orientation} problem, where we are given as input a mixed graph $G=(V,E,A)$ and a set of $k$ demand pairs $(s_i,t_i)$, $i\in[k]$. The goal is to orient the undirected edges of $G$ in a way that the resulting…

Given a simple graph $G$, its line graph, denoted by $L(G)$, is obtained by representing each edge of $G$ as a vertex, with two vertices in $L(G)$ adjacent whenever the corresponding edges in $G$ share a common endpoint. By applying the…

We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an…

A digraph is eulerian if it is connected and every vertex has its in-degree equal to its out-degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs…

For every integer $g$, isomorphism of graphs of Euler genus at most $g$ can be decided in linear time. This improves previously known algorithms whose time complexity is $n^{O(g)}$ (shown in early 1980's), and in fact, this is the first…

For an arbitrary undirected simple graph G with m edges, we give an algorithm with running time O(m^4 |L|^2) to generate the set L of all minimal edge dominating sets of G. For bipartite graphs we obtain a better result; we show that their…

A 2-edge-coloured graph $G$ is {\bf supereulerian} if $G$ contains a spanning closed trail in which the edges alternate in colours. An {\bf eulerian factor} of a 2-edge-coloured graph is a collection of vertex disjoint induced subgraphs…

We study the problem Symmetric Directed Multicut from a parameterized complexity perspective. In this problem, the input is a digraph $D$, a set of cut requests $C=\{(s_1,t_1),\ldots,(s_\ell,t_\ell)\}$ and an integer $k$, and the task is to…

Given an undirected graph $G$, the problem of deciding whether $G$ admits a simple and proper time-labeling that makes it temporally connected is known to be NP-hard (G\"obel et al., 1991). In this article, we relax this problem and ask…

An old conjecture of Bollob\'as and Scott asserts that every Eulerian directed graph with average degree $d$ contains a directed cycle of length at least $\Omega(d)$. The best known lower bound for this problem is $\Omega(d^{1/2})$ by…

A graph $G$ is said to be a $(k,\ell)$-graph if its vertex set can be partitioned into $k$ independent sets and $\ell$ cliques. It is well established that the recognition problem for $(k,\ell)$-graphs is NP-complete whenever $k \geq 3$ or…

The isomorphism problem for graphs (GI) and the isomorphism problem for groups (GrISO) have been studied extensively by researchers. The current best algorithms for both these problems run in quasipolynomial time. In this paper, we study…

We consider the parameterised $k,e$-Long Cycle problem, in which you are given an $n$-vertex undirected graph $G$, a specified edge $e$ in $G$, and a positive integer $k$, and are asked to decide if the graph $G$ has a simple cycle through…