380 research outputs found
Maximum Independent Sets in Subcubic Graphs: New Results
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
New results on word-representable graphs
A graph is word-representable if there exists a word over the
alphabet such that letters and alternate in if and only if
for each . The set of word-representable graphs
generalizes several important and well-studied graph families, such as circle
graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at
most 3, etc. By answering an open question from [M. Halldorsson, S. Kitaev and
A. Pyatkin, Alternation graphs, Lect. Notes Comput. Sci. 6986 (2011) 191--202.
Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in
Computer Science, WG 2011, Tepla Monastery, Czech Republic, June 21-24, 2011.],
in the present paper we show that not all graphs of vertex degree at most 4 are
word-representable. Combining this result with some previously known facts, we
derive that the number of -vertex word-representable graphs is
Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs
We discover new hereditary classes of graphs that are minimal (with respect
to set inclusion) of unbounded clique-width. The new examples include split
permutation graphs and bichain graphs. Each of these classes is characterised
by a finite list of minimal forbidden induced subgraphs. These, therefore,
disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming
that all such minimal classes must be defined by infinitely many forbidden
induced subgraphs.
In the same paper, Daligault, Rao and Thomasse make another conjecture that
every hereditary class of unbounded clique-width must contain a labelled
infinite antichain. We show that the two example classes we consider here
satisfy this conjecture. Indeed, they each contain a canonical labelled
infinite antichain, which leads us to propose a stronger conjecture: that every
hereditary class of graphs that is minimal of unbounded clique-width contains a
canonical labelled infinite antichain.Comment: 17 pages, 7 figure
Independent sets of maximum weight in apple-free graphs
We present the first polynomial-time algorithm to solve the maximum weight independent set problem for apple-free graphs, which is a common generalization of several important classes where the problem can be solved efficiently, such as claw-free graphs, chordal graphs, and cographs. Our solution is based on a combination of two algorithmic techniques (modular decomposition and decomposition by clique separators) and a deep combinatorial analysis of the structure of apple-free graphs. Our algorithm is robust in the sense that it does not require the input graph G to be apple-free; the algorithm either finds an independent set of maximum weight in G or reports that G is not apple-free
Universal graphs and universal permutations
Let be a family of graphs and the set of -vertex graphs in .
A graph containing all graphs from as induced subgraphs is
called -universal for . Moreover, we say that is a proper
-universal graph for if it belongs to . In the present paper, we
construct a proper -universal graph for the class of split permutation
graphs. Our solution includes two ingredients: a proper universal 321-avoiding
permutation and a bijection between 321-avoiding permutations and symmetric
split permutation graphs. The -universal split permutation graph constructed
in this paper has vertices, which means that this construction is
order-optimal.Comment: To appear in Discrete Mathematics, Algorithms and Application
From matchings to independent sets
In 1965, Jack Edmonds proposed his celebrated polynomial-time algorithm to find a maximum matching in a graph. It is well-known that finding a maximum matching in G is equivalent to finding a maximum independent set in the line graph of G. For general graphs, the maximum independent set problem is NP-hard. What makes this problem easy in the class of line graphs and what other restrictions can lead to an efficient solution of the problem? In the present paper, we analyze these and related questions. We also review various techniques that may lead to efficient algorithms for the maximum independent set problem in restricted graph families, with a focus given to augmenting graphs and graph transformations. Both techniques have been used in the solution of Edmonds to the maximum matching problem, i.e. in the solution to the maximum independent set problem in the class of line graphs. We survey various results that exploit these techniques beyond the line graphs
Maximum Independent Sets in Subcubic Graphs: New Results
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
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