Given a coloring f : V (G) → IN of graph G and any subgraph H ⊂ G we define fs(H) = ∑ v∈V (H) f(v). In particular, we denote fs(G) by S(f). The coloring f is called an ICcoloring if for any integer k ∈ [1, S(f)] there is a connected subgraph H ⊂ G such that fs(H) = k. Also, we define the IC-index of G to be M(G) = max{ S(f) : f is an IC-coloring of G }. In this paper we examine some well-known ...

We show that every NP problem is polynomially equivalent to a simple combinatorial problem: the membership problem for a special class of digraphs. These classes are defined by means of shadows (projections) and by finitely many forbidden colored (lifted) subgraphs. Our characterization is motivated by the analysis of syntactical subclasses with the full computational power of NP, which were fi...

We consider max coloring on hereditary graph classes. The problem is defined as follows. Given a graph G = (V, E) and positive node weights w : V → (0,∞), the goal is to find a proper node coloring of G whose color classes C1, C2, . . . , Ck minimize k ∑ i=1 maxv∈Ci w(v). We design a general framework which allows to convert approximation algorithms for standard node coloring into algorithms fo...

A set S  V is a induced -paired dominating set if S is a dominating set of G and the induced subgraph is a perfect matching. The induced paired domination number ip(G) is the minimum cardinality taken over all paired dominating sets in G. The minimum number of colours required to colour all the vertices so that adjacent vertices do not receive the same colour and is denoted by (G). The a...

In this paper, we study the conflict-free coloring of graphs induced by neighborhoods. A coloring of a graph is conflict-free if every vertex has a uniquely colored vertex in its neighborhood. The conflict-free coloring problem is to color the vertices of a graph using the minimum number of colors such that the coloring is conflict-free. We consider both closed neighborhoods, where the neighbor...

This paper analyzes what structural features of graph problems allow efficient parallel algorithms. We survey some parallel algorithms for typical problems on three kinds of graphs, outerplanar graphs, trapezoid graphs and in-tournament graphs. Our results on the shortest path problem, the longest path problem and the maximum flow problem on outerplanar graphs, the minimum-weight connected domi...

A b-coloring of a graph is proper coloring its vertices such that each color class contains vertex adjacent to at least one every other class. The b-chromatic number the largest integer k has with colors. In this work we present lower bounds for vertex-deleted subgraph graph, particularly regarding two important classes, claw-free graphs and chordal graphs. We also get G−{x}, when G large girth.

A set coloring α of a graph G is defined as an assignment of distinct subsets of a finite set X of colors to the vertices of G such that all the colors of the edges which are obtained as the symmetric differences of the sets assigned to their end-vertices are distinct. Additionally, if all the sets on the vertices and edges of G form the set of all nonempty subsets of X, then the coloring α is ...

The concept of induced paired domination number of a graph was introduced by D.S.Studer, T.W. Haynes and L.M. Lawson11, with the following application in mind. In the guard application an induced paired dominating set represents a configuration of security guards in which each guard is assigned one other as a designated backup with in (as in a paired dominating set), but to avoid conflicts (suc...