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 hu...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 158; no. 12; pp. 1268 - 1274
Main Authors: Dourado, Mitre C., Protti, Fábio, Szwarcfiter, Jayme L.
Format: Journal Article Conference Proceeding
Language:English
Published: Kidlington Elsevier B.V 28.06.2010
Elsevier
Subjects:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Combinatorics
Combinatorics. Ordered structures
Computer science; control theory; systems
Convex and discrete geometry
Exact sciences and technology
Geometry
Information retrieval. Graph
m-convex set
m-convexity number
m-hull number
Mathematics
Monophonic convexity
Monophonic number
Sciences and techniques of general use
Theoretical computing
Monophonic number
m-convexity number
m-convex set
Monophonic convexity
m-hull number
Vertex
Convex set
Graph path
Closure
Computer theory
Computing
Computational complexity
Optimization
Polynomial time
Convex hull
Maximum
Convexity
Combinatorics
ISSN:0166-218X, 1872-6771
Online Access:Get full text
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Summary: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.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2009.11.016