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We study a family of problems where the goal is to make a graph Eulerian, i.e., connected and with all the vertices having even degrees, by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions, with or without the requirement of connectivity, etc. The(More)
We study parameterized complexity of a generalization of the classical Feedback Vertex Set problem, namely the Group Feedback Vertex Set problem: we are given a graph $$G$$ G with edges labeled with group elements, and the goal is to compute the smallest set of vertices that hits all cycles of $$G$$ G that evaluate to a non-null element of the group. This(More)
The Pathwidth One Vertex Deletion (POVD) problem asks whether, given an undirected graph G and an integer k, one can delete at most k vertices from G so that the remaining graph has pathwidth at most 1. The question can be considered as a natural variation of the extensively studied Feedback Vertex Set (FVS) problem, where the deletion of at most k vertices(More)
Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves (2011) initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space ℝ l ${{\mathbb(More)
Let η≥0 be an integer and G be a graph. A set X⊆V(G) is called a η-treewidth modulator in G if G∖X has treewidth at most η. Note that a 0-treewidth modulator is a vertex cover, while a 1-treewidth modulator is a feedback vertex set of G. In the η/ρ-Treewidth Modulator problem we are given an undirected graph G, a ρ-treewidth modulator X⊆V(G) in G, and an(More)
We study the Cutwidth problem, where the input is a graph G, and the objective is find a linear layout of the vertices that minimizes the maximum number of edges intersected by any vertical line inserted between two consecutive vertices. We give an algorithm for Cutwidth with running time O(2 k n O(1)). Here k is the size of a minimum vertex cover of the(More)
The Randi´c index R(G) of a graph G is the sum of weights (deg(u) deg(v)) −0.5 over all edges uv of G, where deg(v) denotes the degree of a vertex v. Let r(G) be the radius of G. We prove that for any connected graph G of maximum degree four which is not a path with even number of vertices, R(G) ≥ r(G). As a consequence, we resolve the conjecture R(G) ≥(More)
The 2-Disjoint Connected Subgraphs problem, given a graph along with two disjoint sets of terminals Z 1,Z 2, asks whether it is possible to find disjoint sets A 1,A 2, such that Z 1⊆A 1, Z 2⊆A 2 and A 1,A 2 induce connected subgraphs. While the naive algorithm runs in O(2 n n O(1)) time, solutions with complexity of form O((2−ε) n ) have been found only for(More)
In a scheduling problem, denoted by 1|prec|∑C i in the Graham notation, we are given a set of n jobs, together with their processing times and precedence constraints. The task is to order the jobs so that their total completion time is minimized. 1|prec|∑C i is a special case of the Traveling Repairman Problem with precedences. A natural dynamic programming(More)
A connected graph has tree-depth at most $$k$$ k if it is a subgraph of the closure of a rooted tree whose height is at most $$k$$ k . We give an algorithm which for a given $$n$$ n -vertex graph $$G$$ G , in time $${\mathcal {O}^*}(1.9602^n)$$ O ∗ ( 1 . 9602 n ) computes the tree-depth of $$G$$ G . Our algorithm is based on combinatorial results revealing(More)