Vassilis Giakoumakis

• Professor
• Professor (Full) at University of Picardie Jules Verne

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be NP-complete even for very restricted graph classes such as for claw-free graphs, for...

A graph is hole- and diamond-free (HD-free) if none of its induced subgraphs is isomorphic to a chordless cycle of length at least five or to a diamond. Using the clique separator approach and the simple structure of atoms of HD-free graphs, we show how to recognize HD-free graphs in time . One of the main tools is Lexicographic Breadth-First Searc...

Let $G$ be a finite undirected graph. A vertex {\em dominates} itself and all its neighbors in $G$. A vertex set $D$ is an {\em efficient dominating set} (\emph{e.d.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.\ in $G$, is...

In the normal life span of large enterprises, the strategic management of IT often evolves. Existing services must be replaced with new services without impairing operations. The problem of scheduling such replacement is of critical importance for the success of the operation. We analyze this problem from a quantitative point of view, underlining t...

The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated and most important problems on graphs, it is well known to be NP-complete and hard to approximate. The complexity of MWIS is open for hole-free graphs (i.e., grap...

Clique separator decomposition introduced by Tarjan and Whitesides is one of the most important graph decompositions. A graph is an atom if it has no clique separator. A hole is a chordless cycle with at least five vertices, and an antihole is the complement graph of a hole. A graph is weakly chordal if it is hole-and antihole-free. K 4 − e is also...

We discuss a problem arising in the strategic management of IT enterprises: that of replacing some existing services with new services without impairing operations. We formalize the problem by means of a Mathematical Programming formulation of the Mixed-Integer Nonlinear Programming class and show it can be solved to a satisfactory optimality appro...

We present a linear time recognition algorithm as well as a 4-expression for calculating the clique-width for the co-distance hereditary graphs which is the complementary class of the well known family of distance hereditary graphs.

Let G be a graph, a split in G is a bi-partition (X,Y) of its vertex set V(G) such that |X|,|Y|≥2 and there are all possible edges between X + =X∩N(Y) and Y + =Y∩N(X), where N(X) and N(Y) are respectively neighborhood of X and Y in G. Let X and Y be respectively the sets X∖X + and Y∖Y + . Whenever X - =∅ (resp. Y - =∅) the set X (resp. Y) is a non-...

The substitution composition of two disjoint graphs G1 and G2 is obtained by first removing a vertex x from G2 and then making every vertex in G1 adjacent to all neighbours of x in G2. Let F be a family of graphs defined by a set Z of forbidden configurations. Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathem...

Let H be a graph then a graph H′ is a prime extension of H if H′ is prime (in the sense of modular decomposition), it contains an induced subgraph isomorphic to H and is minimal with respect to set inclusion and primality. An open problem concerning the set of prime extension Ext(H) of H is the following: find the necessary and sufficient condition...

In [7] was introduced a new decomposition scheme for bipartite graphs that was called canonical decomposition. Weak-bisplit graphs are totally decomposable following this decomposition. We give here linear time algorithms for the recognition of weak-bisplit graphs as well as for two subclasses of this class, the P6-free bipartite graphs and the bi-...

We introduced in [4]a new decomposition scheme for bipartite graphs that we called canonical decomposition. Weak bisplit graphs are totally decomposable following this decomposition We give here the frame work for obtaining linear recognition algorithms for Weak bisplit graphs as well as for two sub families of this class the P-6 free bipartite gra...

International audience We study the P_4-tidy graphs, a new class defined by Rusu [30] in order to illustrate the notion of P_4-domination in perfect graphs. This class strictly contains the P_4-extendible graphs and the P_4-lite graphs defined by Jamison & Olariu in [19] and [23] and we show that the P_4-tidy graphs and P_4-lite graphs are closely...

We use the modular decomposition to give O(n(m + n)) algorithms for finding a maximum weighted clique (respectively stable set) and an approximate weighted colouring (respectively partition into cliques) in a (P5 ; P5 )-free graph. As a by-product, we obtain an O(m+n) algorithm for finding a minimum weighted transversal of the C5 in a (P5 ; P5 )- f...

We use the modular decomposition to give O(n(m + n) algorithms for finding a maximum weighted clique (respectively stable set) and an approximate weighted colouring (respectively partition into cliques) in a graph. As a by-product, we obtain an O(m + n) algorithm for finding a minimum weighted transversal of the C5 in a graph.

In this paper we investigate the closure ∗ under substitution-composition of a family of graphs , defined by a set of forbidden configurations. We first prove that ∗ can be defined by a set ∗ of forbidden subgraphs. Next, using a counterexample we show that ∗ is not necessarily a finite set, even when is finite. We then give a sufficient condition...

We use the modular decomposition to give O(n(m + n)) algorithms for finding a maximum weighted clique (respectively stable set) and an approximate weighted colouring (respectively partition into cliques) in a (P5, P5)-free graph. As a by-product, we obtain an O(m+n) algorithm for finding a minimum weighted transversal of the C5 in a (P5, P5)- free...

We study the P4-tidy graphs, a new class defined by Rusu (30) in order to illustrate the notion of P4-domination in perfect graphs. This class strictly contains the P4-extendible graphs and the P4-lite graphs defined by Jamison & Olariu in (19) and (23) and we show that the P4-tidy graphs and P4-lite graphs are closely related. Note that the class...

A graph G was defined in [16] as P4-reducible, if no vertex in G belongs to more than one chordless path on four vertices or P4. A graph G is defined in [15] as P4-sparse if no set of five vertices induces more than one P4, in G. P4-sparse graphs generalize both P4-reducible and the well known class of p4-free graphs or cographs. In an extended abs...

We introduce a new family of bipartite graphs which is the bipartite analogue of the class ofcomplement reduciblegraphs orcographs. Abi-complement reduciblegraph orbi-cographis a bipartite graphG=(W∪B,E) that can be reduced to single vertices by recursively bi-complementing the edge set of all connected bipartite subgraphs. Thebi-complementedgraphḠ...

In this paper, we define a graph G as semi-P-4-sparse if G does not contain as induced subgraph a P-5, a (P-5) over bar or the complement of a fork, where a fork is the tree of order 5 with 3 pendent vertices. This new class of graphs contains strictly the class of P-4-sparse graphs. Using modular decomposition we first propose a linear recognition...

In this paper, we define a graph G as semi-P4-sparse if G does not contain as induced subgraph a P5, a P̄5 or the complement of a fork, where a fork is the tree of order 5 with 3 pendent vertices. This new class of graphs contains strictly the class of P4-sparse graphs. Using modular decomposition we first propose a linear recognition algorithm for...

The scattering number of a graph G equals max {c(G⧹S) − |S|S is a cutset of G} where c(G⧹S) denotes the number of connected components in G⧹S. Jung (1978) has given for any graph having no induced path on four vertices (P4-free graph) a correspondence between the value of its scattering number and the existence of Hamiltonian paths or Hamiltonian c...

We present a new family of graphs, the family of P4-laden graphs strictly containing the class of P4-lite graphs introduced in (Jamison and Olariu, 1989). We first show that P4-laden graphs are brittle and next, using modular decomposition we present for this class of graphs a linear recognition algorithm as well as linear algorithms for classical...

We extend results due to Blázsik et al. (1993) on graphs with no induced C4 and 2K2 to the self-complementary class of ( graphs. Moreover, we obtain an O(ω2) γ-binding function for this last class of graphs, answering thus partially a question of A. Gyárfás.

We show that the atoms by clique separator decomposition of a diamond-free hole-free graph are of three simple types: clique, matched co-bipartite or chordal bipartite. We present an O(n 2) time algorithm to recognize these graphs and decompose them into atoms. To do this, we use algorithm LexBFS in a novel fashion. Given the atoms, we show how to...