Nícolas A. Martins

The b-chromatic index of graphs

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to at least one vertex in each other color class. The b-chromatic number of G is the maximum integer b ( G ) for which G has a b-coloring with b ( G ) colors. This problem was introduced by Irving and Manlove (1999), where they showed that computing b ( G ) is NP -hard in general and polynomial-time solvable for trees. A natural question that arises is whether the edge version of this problem is also NP -hard or not. Here, we prove that computing the b-chromatic index of a graph G is NP -hard, even if G is either a comparability graph or a C k -free graph, and give partial results on the complexity of the problem restricted to trees, more specifically, we solve the problem for caterpillars graphs. Although solving problems on caterpillar graphs is usually quite simple, this problem revealed itself to be unusually hard. The presented algorithm uses a dynamic programming approach that combines partial solutions which are proved to exist if, and only if, a particular polyhedron is non-empty.

b-chromatic index of graphs

Abstract A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to a vertex in each other color class. The b-chromatic number of G is the maximum integer χ b ( G ) for which G has a b-coloring with χ b ( G ) colors. This problem was introduced by Irving and Manlove in 1999, where they showed that computing χ b ( G ) is NP -hard in general and polynomial-time solvable for trees. A natural question that arises is whether the edge version of this problem is also NP -hard or not. Here, we prove that computing the b-chromatic index of a graph G is NP -hard, even if G is either a comparability graph or a C k -free graph, and give some partial results on the complexity of the problem restricted to trees.

Restricted coloring problems on graphs with few P4′s

Abstract In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number and the harmonious chromatic number of P 4 -tidy graphs and ( q , q − 4 )-graphs, for every fixed q. These classes include cographs, P 4 -sparse and P 4 -lite graphs. We also obtain a polynomial time algorithm to determine the Grundy number of ( q , q − 4 )-graphs. All these coloring problems are known to be NP-hard for general graphs.

Spy-Game on graphs

We define and study the following two-player game on a graph G. Let k in N^*. A set of k guards is occupying some vertices of G while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where s in N^* is his speed. Then, each guard may move along one edge. The spy and the guards may occupy same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d in N (a predefined distance) from every guard. Can the spy win against k guards? Similarly, what is the minimum distance d such that k guards may ensure that at least one of them remains at distance at most d from the spy? This game generalizes two well-studied games: Cops and robber games (when s=1) and Eternal Dominating Set (when s is unbounded). We consider the computational complexity of the problem, showing that it is NP-hard and that it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards and the required distance d when G is a path or a cycle. Our main result is that there exists beta>0 such that Omega(n^{1+beta}) guards are required to win in any n*n grid.

Spy-game on graphs: Complexity and simple topologies

We define and study the following two-player game on a graph G. Let k ∈ N *. A set of k guards is occupying some vertices of G while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where s ∈ N * is his speed. Then, each guard may move along one edge. The spy and the guards may occupy the same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d ∈ N (a predefined distance) from every guard. Can the spy win against k guards? Similarly, what is the minimum distance d such that k guards may ensure that at least one of them remains at distance at most d from the spy? This game generalizes two well-studied games: Cops and robber games (when s = 1) and Eternal Dominating Set (when s is unbounded). We consider the computational complexity of the problem, showing that it is NP-hard (for every speed s and distance d) and that some variant of it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards, the speed s of the spy and the required distance d when G is a path or a cycle.

Locally identifying coloring of graphs with few P4s

Abstract A lid-coloring (locally identifying coloring) of a graph is a proper coloring such that, for any edge uv , if u and v have distinct closed neighborhoods, then the set of colors used on vertices of the closed neighborhoods of u and v are distinct. The lid-chromatic number is the minimum number of colors used in a lid-coloring. In this paper we prove a relation between lid-coloring and a variation, called strong lid-coloring. With this, we obtain linear time algorithms to calculate the lid-chromatic number for some classes of graphs with few P 4 s, such as cographs, P 4 -sparse graphs and ( q , q − 4 ) -graphs. We also prove that the lid-chromatic number is O ( n 1 − e ) -inapproximable in polynomial time for every e > 0 , unless P = NP .

L ( 2 , 1 ) -labelling of graphs with few P 4 ’s

Abstract Given a simple graph G , an L ( 2 , 1 ) -labelling (or λ -labelling) of G is a function c : V ( G ) → N such that | c ( x ) − c ( y ) | ≥ 2 , if x and y are neighbors and | c ( x ) − c ( y ) | ≥ 1 if x and y have a common neighbor. The span of a labelling is the difference of the smallest and largest labels used. An L ( 2 , 1 ) -span of a graph G , denoted by λ ( G ) , is a minimum span over all L ( 2 , 1 ) -labellings of G . The problem of determining if λ ( G ) ≤ k is NP-Complete for any k ≥ 4 . In this paper, we obtain a linear time algorithm to compute λ ( G ) for any ( q , q − 4 ) -graph with q fixed. Another important topic regarding the λ -labelling is to bound the λ -chromatic number of a graph by some function of it. Griggs and Yeh conjectured that λ ( G ) ≤ Δ 2 for any graph G with maximum degree Δ ≥ 2 . They also proved that the greedy algorithm for the problem uses at most Δ 2 + Δ . Furthermore we prove that the Griggs–Yeh conjecture is true for P 4 -sparse graphs, P 4 -laden graphs and all ( q , q − 4 ) -graphs with at least 3 q / 2 vertices.

Inapproximability of the lid-chromatic number

A lid-coloring (locally identifying coloring) of a graph is a proper coloring such that, for any edge uv where u and v have distinct closed neighborhoods, the set of colors used on vertices of the closed neighborhoods of u and v are also distinct. In this paper we obtain a relation between lid-coloring and a variation, called strong lid-coloring. With this, we obtain linear time algorithms to calculate the lid-chromatic number for some classes of graphs with few P4s. We also prove that the lid-chromatic number is O(n1/2−e)-inapproximable in polinomial time for every e>0, unless P=NP.