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Communication Dans Un Congrès Lecture Notes in Computer Science Année : 2012

On graph identification problems and the special case of identifying vertices using paths

Résumé

In this paper, we introduce the identifying path cover problem: an \emph{identifying path cover} of a graph $G$ is a set $\mathcal P$ of paths such that each vertex belongs to a path of $\mathcal P$, and for each pair $u,v$ of vertices, there is a path of $\mathcal P$ which includes exactly one of $u,v$. This problem is related to a large variety of identification problems. We investigate the identifying path cover problem in some families of graphs. In particular, we derive the optimal size of an identifying path cover for paths, cycles, hypercubes and topologically irreducible trees and give an upper bound for all 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 $\left\lceil\frac{2(|V(G)|-1)}{3}\right\rceil$. We also 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 a fixed value, the problem is APX-complete.
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Dates et versions

hal-00960561 , version 1 (18-03-2014)

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Florent Foucaud, Matjaž Kovše. On graph identification problems and the special case of identifying vertices using paths. International Workshop on Combinatorial Algorithms (IWOCA), Jul 2012, Krishnankoil, India. pp.32-45, ⟨10.1007/978-3-642-35926-2_4⟩. ⟨hal-00960561⟩

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