Results 1 
7 of
7
Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms
, 2006
"... We consider the concepts of a ttotal vertex cover and a ttotal edge cover (t 1), which generalize the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
We consider the concepts of a ttotal vertex cover and a ttotal edge cover (t 1), which generalize the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has least t vertices (edges). These denitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present NPcompleteness and approximability results (both upper and lower bounds) and FPT algorithms for problems concerned with nding the minimum size of a ttotal vertex cover, ttotal edge cover and connected vertex cover, in particular improving on a previous FPT algorithm for the latter problem.
The Price of Connectivity for Vertex Cover
, 2014
"... The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover n ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the Price of Connectivity, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number t. We obtain forbidden induced subgraph characterizations for every real value t ≤ 3/2. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most t. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NPhard. In fact, we show that it is even complete for the class ΘP2 = PNP [log], the class of decision problems that can be solved in polynomial time, provided we can make O(logn) queries to an NPoracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.
A Note on Connected Dominating Set in Graphs Without Long Paths And Cycles
, 2014
"... ar ..."
(Show Context)
The price of connectivity for feedback vertex set. In:
 Eurocomb 2013, CRM Series,
, 2013
"... Abstract. Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. In general graphs, the ratio cfvs(G)/fvs(G) can be arbitrarily large. We study the interdependence between fvs(G) and cfvs(G) in graph c ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. In general graphs, the ratio cfvs(G)/fvs(G) can be arbitrarily large. We study the interdependence between fvs(G) and cfvs(G) in graph classes defined by excluding one induced subgraph H.
Forbidden Induced Subgraphs and the Price of Connectivity for Feedback Vertex Set
"... Abstract. Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. For a graph class G, the price of connectivity for feedback vertex set (pocfvs) for G is defined as the maximum ratio cfvs(G)/fvs(G) ov ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. For a graph class G, the price of connectivity for feedback vertex set (pocfvs) for G is defined as the maximum ratio cfvs(G)/fvs(G) over all connected graphs G in G. The pocfvs for general graphs is unbounded, as the ratio cfvs(G)/fvs(G) can be arbitrarily large. We study the pocfvs for graph classes defined by a finite family H of forbidden induced subgraphs. We characterize exactly those finite families H for which the pocfvs for Hfree graphs is bounded by a constant. Prior to our work, such a result was only known for the case where H = 1.
The Price of Connectivity for Vertex Cover
"... The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover ..."
Abstract
 Add to MetaCart
The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the Price of Connectivity, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number t. We obtain forbidden induced subgraph characterizations for every real value t ≤ 3/2. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most t. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NPhard. In fact, we show that it is even complete for the class ΘP2 = PNP [log], the class of decision problems that can be solved in polynomial time, provided we can make O(logn) queries to an NPoracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.
On price of symmetrisation
, 2013
"... Abstract. We introduce the price of symmetrisation, a concept that aims to compare fundamental differences (gap and quotient) between values of a given graph invariant for digraphs and the values of the same invariant of the symmetric versions of these digraphs. Basically, given some invariant our ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We introduce the price of symmetrisation, a concept that aims to compare fundamental differences (gap and quotient) between values of a given graph invariant for digraphs and the values of the same invariant of the symmetric versions of these digraphs. Basically, given some invariant our goal is to characterise digraphs that maximise price of symmetrisation. In particular, we show that for some invariants, as diameter or domination number, the problem is easy. The main contribution of this paper is about (partial) results on the price of symmetrisation of the average distance. It appears to be much more intricate than the simple cases mentioned above. First, we state a conjecture about digraphs that maximise this price of symmetrisation. Then, we prove that this conjecture is true for some particular class of digraphs (called bags) but it remains open for general digraphs. Moreover, we study several graph transformations in order to remove some configurations that do not appear in the conjectured extremal digraphs.