Mitre Costa Dourado

Complexity results related to monophonic convexity

The study of monophonic convexity is based on the family of induced paths of a graph. The closure of a subset X of vertices, in this case, contains every vertex v such that v belongs to some induced path linking two vertices of X. Such a closure is called monophonic closure. Likewise, the convex hull of a subset is called monophonic convex hull. In this work we deal with the computational complexity of determining important convexity parameters, considered in the context of monophonic convexity. Given a graph G, we focus on three parameters: the size of a maximum proper convex subset of G (m-convexity number); the size of a minimum subset whose closure is equal to V(G) (monophonic number); and the size of a minimum subset whose convex hull is equal to V(G) (m-hull number). We prove that the decision problems corresponding to the m-convexity and monophonic numbers are NP-complete, and we describe a polynomial time algorithm for computing the m-hull number of an arbitrary graph.

Some remarks on the geodetic number of a graph

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G. We prove that it is NP-complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs.

On the Hull Number of Triangle-Free Graphs

A set of vertices

On the Carathéodory Number for the Convexity of Paths of Order Three

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More fires and more fighters

in a graph is convex if it contains all vertices which lie on shortest paths between vertices in

The maximum time of 2-neighbour bootstrap percolation

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Algorithmic Aspects of Monophonic Convexity

. The convex hull of a set of vertices

Computational aspects of the Helly property: a survey

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On the Carathéodory number of interval and graph convexities

is the smallest convex set containing

Helly property, clique graphs, complementary graph classes, and sandwich problems

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