Nicola Apollonio

The maximum vertex coverage problem on bipartite graphs

Given a simple undirected graph G and a positive integer s the Maximum Vertex Coverage Problem is the problem of finding a set U of s vertices of G such that the number of edges having at least one endpoint in U is as large as possible. We prove that the Maximum Vertex Coverage problem on bipartite graphs is NP-hard and discuss several consequences related to known combinatorial optimization problems.

A characterization of partial directed line graphs

Can a directed graph be completed to a directed line graph? If possible, how many arcs must be added? In this paper we address the above questions characterizing partial directed line (PDL) graphs, i.e., partial subgraph of directed line graphs. We show that for such class of graphs a forbidden configuration criterion and a Krauszs like theorem are equivalent characterizations. Furthermore, the latter leads to a recognition algorithm that requires O(m) worst case time, where m is the number of arcs in the graph. Given a partial line digraph, our characterization allows us to find a minimum completion to a directed line graph within the same time bound. The class of PDL graphs properly contains the class of directed line graphs, characterized in [J. Blazewicz, A. Hertz, D. Kobler, D. de Werra, On some properties of DNA graphs, Discrete Appl. Math. 98(1-2) (1999) 1-19], hence our results generalize those already known for directed line graphs. In the undirected case, we show that finding a minimum line graph edge completion is NP-hard, while the problem of deciding whether or not an undirected graph is a partial graph of a simple line graph is trivial.

Minsquare Factors and Maxfix Covers of Graphs

We provide a polynomial algorithm that determines for any given undirected graph, positive integer k and various objective functions on the edges or on the degree sequences, as input, k edges that minimize the given objective function. The tractable objective functions include linear, sum of squares, etc. The source of our motivation and at the same time our main application is a subset of k vertices in a line graph, that cover as many edges as possible (maxfix cover). Besides the general algorithm and connections to other problems, the extension of the usual improving paths for graph factors could be interesting in itself: the objects that take the role of the improving walks for b-matchings or other general factorization problems turn out to be edge-disjoint unions of pairs of alternating walks. The algorithm we suggest also works if for any subset of vertices upper, lower bound constraints or parity constraints are given. In particular maximum (or minimum) weight b-matchings of given size can be determined in polynomial time, combinatorially, in more than one way.

Note: Integrality properties of edge path tree families

An Edge Path Tree (EPT) family is a family whose members are edge sets of paths in a tree. Relying on the notion of Pie introduced in [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinatorial Theory, Series B 38 (1985) 8-22], we characterize Ideal and Mengerian EPT families. In particular, we show that an EPT family is Ideal if and only if it is Mengerian. If, in addition, the EPT family is uniform, then it is Ideal if and only if it is Unimodular. The latter equivalence generalizes the well-known fact that the edge set of a graph is an Ideal clutter if and only if the graph is bipartite.

On the Galois lattice of bipartite distance hereditary graphs

We give a complete characterization of bipartite graphs having tree-like Galois lattices. We prove that the poset obtained by deleting bottom and top elements from the Galois lattice of a bipartite graph is tree-like if and only if the graph is a bipartite distance hereditary graph. Relations with the class of Ptolemaic graphs are discussed and exploited to give an alternative proof of the result.

Improved Approximation of Maximum Vertex Coverage Problem on Bipartite Graphs

Given a simple undirected graph

Recognizing Helly Edge-Path-Tree graphs and their clique graphs

G

A superclass of Edge-Path-Tree graphs with few cliques

and a positive integer

The Sunfish Against the Octopus: Opposing Compactness to Gerrymandering

s

Cardinality constrained path covering problems in grid graphs

, the maximum vertex coverage problem (MVC) is the problem of finding a set