Identifying path covers in graphs

  • Authors:

  • Florent Foucaud;Matja Kovše

  • Affiliations:

  • Univ. Bordeaux, LaBRI, UMR5800, F-33400 Talence, France and CNRS, LaBRI, UMR5800, F-33400 Talence, France;Bioinformatics Group, Department of Computer Science and Interdisciplinary Center for Bioinformatics, Univ. of Leipzig, Härtelstrasse 16-18, D-04107 Leipzig, Germany

  • Venue:
  • Journal of Discrete Algorithms
  • Year:
  • 2013

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Abstract

This paper introduces the problem of identifying vertices of a graph using paths. An identifying path cover of a graph G is a set P of paths such that each vertex belongs to a path of P, and for each pair u, v of vertices, there is a path of P which includes exactly one of u, v. This new notion is related to a large number of other identification problems in graphs and hypergraphs. We study the identifying path cover problem under both combinatorial and algorithmic points of view. In particular, we derive the optimal size of an identifying path cover for paths, cycles and hypercubes, and give upper bounds for trees. We give lower and upper bounds on the minimum size of an identifying path cover for general graphs, and discuss their tightness. In particular, we show that any connected graph G has an identifying path cover of size at most @?2|V(G)|3@?+5. We then study the computational complexity of the associated optimization problem, in particular we show that when the length of the paths is asked to be of bounded value, the problem is APX-complete.