On rigid circuit graphs

Dirac, G. A.

doi:10.1007/bf02992776

Cited by 803 publications

(481 citation statements)

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“…This also implies that every minimal separator is a clique [9]. It is also important to note that chordal graphs have at most n maximal cliques [9], and hence at most n − 1 minimal separators. Chordal graphs can be recognized and clique trees can be computed in linear time [15].…”

Section: Proposition 4 For a Potential Maximal Clique ω Of The Graphmentioning

confidence: 84%

A Parameterized Algorithm for Chordal Sandwich

Heggernes

Mancini

Nederlof

et al. 2010

Lecture Notes in Computer Science

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Given an arbitrary graph G = (V, E) and an additional set of admissible edges F , the Chordal Sandwich problem asks whether there exists a chordal graph (V, E ∪ F ) such that F ⊆ F . This problem arises from perfect phylogeny in evolution and from sparse matrix computations in numerical analysis, and it generalizes the widely studied problems of completions and deletions of arbitrary graphs into chordal graphs. As many related problems, Chordal Sandwich is NP-complete. In this paper we show that the problem becomes tractable when parameterized with a suitable natural measure on the set of admissible edges F . In particular, we give an algorithm with running time O(2 k n 5 ) to solve this problem, where k is the size of a minimum vertex cover of the graph (V, F ). Hence we show that the problem is fixed parameter tractable when parameterized by k. Note that the parameter does not assume any restriction on the input graph, and it concerns only the additional edge set F .

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Section: Proposition 4 For a Potential Maximal Clique ω Of The Graphmentioning

confidence: 84%

A Parameterized Algorithm for Chordal Sandwich

Heggernes

Mancini

Nederlof

et al. 2010

Lecture Notes in Computer Science

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“…To color the interference graph we employ the fact that there is a perfect elimination order (PEO) for each chordal graph [7]. A PEO defines an ordering < of nodes of a graph G, such that each successively removed node is simplicial, which means it forms a clique with all remaining neighbors in G. After spilling, assigning the registers in reverse PEO ensures that each node is simplicial and thus has a free register available.…”

Section: Register Allocation On Ssa Formmentioning

confidence: 99%

SSA-Based Register Allocation with PBQP

Buchwald

Zwinkau

Bersch

2011

Lecture Notes in Computer Science

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Abstract. Recent research shows that maintaining SSA form allows to split register allocation into separate phases: spilling, register assignment and copy coalescing. After spilling, register assignment can be done in polynomial time, but copy coalescing is NP-complete. In this paper we present an assignment approach with integrated copy coalescing, which maps the problem to the Partitioned Boolean Quadratic Problem (PBQP). Compared to the state-of-the-art recoloring approach, this reduces the relative number of swap and copy instructions for the SPEC CINT2000 benchmark to 99.6% and 95.2%, respectively, while taking 19% less time for assignment and coalescing.

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“…A graph is chordal if it contains no hole and no square as an induced subgraph. By a theorem of Dirac [9], every chordal graph contains a simplicial vertex. So, the simplicial vertex reduction can be used to determine the stability number of a chordal graph.…”

Section: Simplicial Neighborhood and Magnet Reductionsmentioning

confidence: 99%

Graph transformations preserving the stability number

Lévêque

Werra

2009

Electronic Notes in Discrete Mathematics

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In this note we analyse the relations between several graph transformations that were introduced to be used in procedures determining the stability number of a graph. We show that all these transformations can be decomposed into a sequence of edge deletions and twin deletions. We also show how some of these transformations are related to the notion of even pair introduced to color some classes of perfect graphs. Then, some properties of edge deletion and twin deletion are given and a conjecture is formulated about the class of graphs for which these transformations can be used to determine the stability number.

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On rigid circuit graphs

Cited by 803 publications

References 1 publication

A Parameterized Algorithm for Chordal Sandwich

A Parameterized Algorithm for Chordal Sandwich

SSA-Based Register Allocation with PBQP

Graph transformations preserving the stability number

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