Ugo Pietropaoli

Hitting diamonds and growing cacti

We consider the following NP-hard problem: in a weighted graph, find a minimum cost set of vertices whose removal leaves a graph in which no two cycles share an edge. We obtain a constant-factor approximation algorithm, based on the primal-dual method. Moreover, we show that the integrality gap of the natural LP relaxation of the problem is Θ(logn), where n denotes the number of vertices in the graph.

On the recognition of fuzzy circular interval graphs

Fuzzy circular interval graphs are a generalization of proper circular arc graphs and have been recently introduced by Chudnovsky and Seymour as a fundamental subclass of claw-free graphs. In this paper, we provide a polynomial time algorithm for recognizing such graphs, and more importantly for building a suitable model for these graphs.

Vertex-colouring of 3-chromatic circulant graphs

Abstract A circulant graph C n ( a 1 , … , a k ) is a graph with n vertices { v 0 , … , v n − 1 } such that each vertex v i is adjacent to vertices v ( i + a j ) m o d n , for j = 1 , … , k . In this paper we investigate the vertex colouring problem on circulant graphs. We approach the problem in a purely combinatorial way based on an array representation and propose an exact O ( k 3 log 2 n + n ) algorithm for a subclass of 3-chromatic C n ( a 1 , … , a k ) ’s with k ≥ 2 , which are characterized in the paper.

Coloring Toeplitz graphs

Abstract Let n , a 1 , a 2 , … , a k be distinct positive integers. A finite Toeplitz graph T n ( a 1 , a 2 , … , a k ) = ( V , E ) is a graph where V = { v 0 , v 1 , … , v n − 1 } and E = { ( v i , v j ) , for | i − j | ∈ { a 1 , a 2 , … , a k } } . If the number of vertices is infinite, we get an infinite Toeplitz graph. In this paper we first give a complete characterization for connected bipartite finite/infinite Toeplitz graphs. We then focus on finite/infinite Toeplitz graphs with k ⩽ 3 , and provide a characterization of their chromatic number.