Phablo F. S. Moura

Polyhedral studies on the convex recoloring problem

Abstract A coloring of the vertices of a graph G is convex if, for each assigned color d, the vertices with color d induce a connected subgraph of G. We address the convex recoloring problem, defined as follows. Given a graph G and a coloring of its vertices, recolor a minimum number of vertices of G, so that the resulting coloring is convex. This problem is known to be NP-hard even when G is a path. We show an integer programming formulation for the weighted version of this problem on arbitrary graphs, and then specialize it for trees. We study the facial structure of the polytope defined as the convex hull of the integer points satisfying the restrictions of the proposed ILP formulation, present several classes of facet-defining inequalities and discuss separation algorithms.

On the proper orientation number of bipartite graphs

An orientation of a graph G is a digraph D obtained from G by replacing each edge by exactly one of the two possible arcs with the same endvertices. For each v ? V ( G ) , the indegree of v in D, denoted by d D - ( v ) , is the number of arcs with head v in D. An orientation D of G is proper if d D - ( u ) ? d D - ( v ) , for all u v ? E ( G ) . The proper orientation number of a graph G, denoted by ? ? ( G ) , is the minimum of the maximum indegree over all its proper orientations. In this paper, we prove that ? ? ( G ) ? ( Δ ( G ) + Δ ( G ) ) / 2 + 1 if G is a bipartite graph, and ? ? ( G ) ? 4 if G is a tree. It is well-known that ? ? ( G ) ? Δ ( G ) , for every graph G. However, we prove that deciding whether ? ? ( G ) ? Δ ( G ) - 1 is already an NP -complete problem on graphs with Δ ( G ) = k , for every k ? 3 . We also show that it is NP -complete to decide whether ? ? ( G ) ? 2 , for planar subcubic graphs G. Moreover, we prove that it is NP -complete to decide whether ? ? ( G ) ? 3 , for planar bipartite graphs G with maximum degree 5.

The k-hop connected dominating set problem: hardness and polyhedra☆

Abstract Let G = ( V , E ) be a connected graph, and k a positive integer. A subset D ⊆ V is a k-hop connected dominating set (k-CDS) if the subgraph of G induced by D is connected and, for every vertex v in G, there is a vertex u in D such that the distance between v and u is at most k. We study the problem of finding a minimum k-hop connected dominating set, denoted by the acronym Min k-CDS. Firstly, we prove that Min k-CDS is NP -hard on planar bipartite graphs of maximum degree 4 and on planar biconnected graphs of maximum degree 5. We present an inapproximability threshold for Min k-CDS on bipartite and on (1, 2)-split graphs, and we also prove that Min k-CDS is APX -hard on bipartite graphs of maximum degree 4. These results are shown to hold for every positive integer k. For k = 1 , the classical minimum connected dominating set problem, we present an integer linear programming formulation and show some classes of inequalities that define facets of the corresponding polytope. We also present an approximation algorithm for this case.

On the representatives k-fold coloring polytope

Abstract A k -fold x -coloring of a graph G is an assignment of (at least) k distinct colors from the set { 1 , 2 , … , x } to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The k -th chromatic number of G , denoted by χ k ( G ) , is the smallest x such that G admits a k -fold x -coloring. We present an ILP formulation to determine χ k ( G ) and study the facial structure of the corresponding polytope P k ( G ) . We show facets that P k + 1 ( G ) inherits from P k ( G ) . We also relate P k ( G ) to P 1 ( G ∘ K k ) , where G ∘ K k is the lexicographic product of G by a clique with k vertices. In both cases, we can obtain facet-defining inequalities from many of those known for the 1-fold coloring polytope. In addition, we present a class of facet-defining inequalities based on strongly χ k -critical webs, which extend and generalize known corresponding results for 1-fold coloring. We introduce this criticality concept and characerize the webs having such a property.

Strong intractability of generalized convex recoloring problems

Abstract A coloring of the vertices of a connected graph is r-convex if each color class induces a subgraph with at most r components. We address the r-convex recoloring problem defined as follows. Given a graph G and a coloring of its vertices, recolor a minimum number of vertices of G so that the resulting coloring is r-convex. This problem, known to be NP -hard even on paths, was first investigated on trees and for r = 1 , motivated by applications on perfect phylogenies. The concept of r-convexity, for r ≥ 2 , was proposed later, and it is also of interest in the study of protein-protein interaction networks and phylogenetic networks. Here, we show that, for each r ∈ N , the r-convex recoloring problem on n-vertex bipartite graphs cannot be approximated within a factor of n 1 − e for any e > 0 , unless P = NP . We also provide strong hardness results for weighted and parametrized versions of the problem.

Lifted, projected and subgraph-induced inequalities for the representatives k -fold coloring polytope

A k -fold x -coloring of a graph G is an assignment of (at least) k distinct colors from the set { 1 , 2 , ź , x } to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The k th chromatic number ofź G , denoted byź ź k ( G ) , is the smallest x such that G admits a k -fold x -coloring. We present an integer linear programming formulation (ILP) to determine ź k ( G ) and study the facial structure of the corresponding polytope P k ( G ) . We show facets thatź P k + 1 ( G ) inherits from P k ( G ) and show how to lift facets from P k ( G ) to P k + ź ( G ) . We project facets of P 1 ( G ź K k ) into facets of P k ( G ) , where G ź K k is the lexicographic product of G by a clique with k vertices. In both cases, we can obtain facet-defining inequalities from many of those known for the 1 -fold coloring polytope. We also derive facets of P k ( G ) from facets of stable set polytopes of subgraphs of G . In addition, we present classes of facet-defining inequalities based on strongly ź k -critical webs and antiwebs, which extend and generalize known results for 1 -fold coloring. We introduce this criticality concept and characterize the webs and antiwebs having such a property.