A tree t-spanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The Tree t-Spanner problem asks whether a graph admits a tree t-spanner, given t. We substantially strengthen the hardness result of Cai and Corneil [SIAM J. Discrete Math. 8 (1995) 359– 387] by showing that, for any t ≥ 4, Tree t-Spanner is ...

A graph is unipolar if it can be partitioned into a clique and a disjoint union of cliques, and a graph is a generalised split graph if it or its complement is unipolar. A unipolar partition of a graph can be used to find efficiently the clique number, the stability number, the chromatic number, and to solve other problems that are hard for general graphs. We present an O(n)-time algorithm for ...

A set D ⊆ V of a graph G = (V,E) is called a restrained dominating set of G if every vertex not in D is adjacent to a vertex in D and to a vertex in V \D. The MINIMUM RESTRAINED DOMINATION problem is to find a restrained dominating set of minimum cardinality. Given a graph G, and a positive integer k, the RESTRAINED DOMINATION DECISION problem is to decide whether G has a restrained dominating ...

We consider the following vertex-partition problem on graphs: given a graph with real nonnegative edge weights, partition the vertices into clusters (in this case cliques) to minimize the total weight of edges out of the clusters. This optimization problem is known to be an NP-complete problem even for unweighted graphs and has been studied extensively in the scope of fixed-parameter tractabili...

In this paper, we define the notions of inverse strong non-split r-dominating set and inverse strong non-split r-domination number γ′snsr(G) of a graph G. We characterize graphs for which γsnsr(G) + γ′snsr(G) = n, where γsnsr(G) is the strong non-split r-domination number of G. We get many bounds on γ′snsr(G). Nordhaus-Gaddum type results are also obtained for this new parameter.

We revisit in this paper the probabilistic coloring problem ( ) and focus ourselves on bipartite and split graphs. We first give some general properties dealing with the optimal solution. We then show that the unique 2-coloring achieves approximation ratio 2 in bipartite graphs under any system of vertex-probabilities and propose a polynomial algorithm achieving tight appro...

A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. A Chord...

Chordal graph shelling antimatroids have received little attention with regard to their combinatorial properties and related optimization problems, as compared to the case of poset shelling antimatroids. Here we consider a special case of these antimatroids, namely the split graph shelling antimatroids. We show that the feasible sets of such an antimatroid relate to some poset shelling antimatr...

The square of a graph G, denoted by G, is obtained from G by putting an edge between two distinct vertices whenever their distance is two. Then G is called a square root of G. Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a split graph, that is, a graph in which the vertex set can be partitioned into a stable set and a clique. We...

The problem of maintaining a representation of a dynamic graph as long as a certain property is satisfied, has recently been considered for a number of properties. This paper presents an optimal algorithm for this problem on vertex-dynamic connected distance hereditary graphs: both vertex insertion and deletion have complexity O(d), where d is the degree of the vertex involved in the modificati...