The maximum independent set problem is known to be NP-hard in the class of
subcubic graphs, i.e. graphs of vertex degree at most 3. We present a
polynomial-time solution in a subclass of subcubic graphs generalizing several
previously known results

A Brandstädt

F Gavril

G Alexe

GJ Minty

HL Bodlaender

OJ Murphy

Robert E. Tarjan

S Whitesides

VV Lozin

English

arXiv

International audienceWe consider the complexity of the classical Independent Set problem on classes of subcubic graphs characterized by a finite set of forbidden induced subgraphs. It is well-known that a necessary condition for Independent Set to be tractable in such a class (unless P=NP) is that the set of forbidden induced subgraphs includes a subdivided star S k,k,k , for some k. Here, S k,k,k is the graph obtained by taking three paths of length k and identifying one of their endpoints. It is an interesting open question whether this condition is also sufficient: is Independent Set tractable on all hereditary classes of subcu-bic graphs that exclude some S k,k,k ? A positive answer to this question would provide a complete classification of the complexity of Independent Set on all classes of subcubic graphs characterized by a finite set of forbidden induced subgraphs. The best currently known result of this type is tractability for S2,2,2-free graphs. In this paper we generalize this result by showing that the problem remains tractable on S 2,k,k-free graphs, for any fixed k. Along the way, we show that subcubic Independent Set is tractable for graphs excluding a type of graph we call an "apple with a long stem", generalizing known results for apple-free graphs

Harutyunyan, Ararat

Lampis, Michael

Lozin, Vadim

Monnot, Jérôme

Archive Ouverte en Sciences de l'Information et de la Communication

Maximum Independent Sets in Subcubic Graphs: New Results

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The maximum independent set problem is known to be NP-hard in the class of subcubic graphs, i.e. graphs of vertex degree at most 3. We study complexity of the problem on hereditary subclasses of subcubic graphs. Each such subclass can be described by means of forbidden induced subgraphs. In case of finitely many forbidden induced subgraphs a necessary condition for polynomial-time solvability of the problem in subcubic graphs (unless ) is the exclusion of the graph , which is a tree with three leaves of distance  from the only vertex of degree 3. Whether this condition is also sufficient is an open question, which was previously answered only for -free subcubic graphs and -free subcubic graphs. Combining various algorithmic techniques, in the present paper we generalize both results and show that the problem can be solved in polynomial time for -free subcubic graphs, for any fixed value of k

Maximum Independent Sets in Subcubic Graphs: New Results

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