Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width

Courcelle, Bruno; Makowsky, Johann A.; Rotics, Udi

doi:10.1007/s002249910009

Cited by 694 publications

(459 citation statements)

References 35 publications

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“…If there is a nonsimplicial vertex h of π(h) representing a non-complete graph, then G[h] is a starfish or an urchin and h is a vertex of the body representing a non-complete graph; if so, Theorem 2.4 implies that h represents 2K 1 and the algorithm correctly sets C(h) to True. Hence, we if h has at least three nonleaf children then 10 output "G is not neighborhood-perfect" and stop 11 else if h has exactly two nonleaf children h1 and h2 and at least one of C(h1) and C(h2) is True then 12 output "G is not neighborhood-perfect" and stop 13 Step 2: 14 output "G is neighborhood-perfect" assume without loss of generality, that every nonsimplicial vertex of π(h) represents a complete graph. We conclude that each induced C 4 of G[h] arises from two adjacent simplicial vertices h 1 and h 2 of π(h), each of which represents a non-complete graph.…”

Section: Recognition Algorithmsmentioning

confidence: 99%

“…It is worth noting that a different approach for obtaining linear-time algorithms for P 4 -tidy graphs (and, more generally, in graph classes having bounded clique-width) was introduced by Courcelle et al in [11]. This approach allows for linear-time solutions of recognition and optimization problems that are expressible in a certain monadic second-order logic.…”

Section: Further Remarksmentioning

confidence: 99%

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Neighborhood covering and independence on $$P_4$$P4-tidy graphs and tree-cographs

2017

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Given a simple graph G, a set C ⊆ V (G) is a neighborhood cover set if every edge and vertex of G belongs to someLet ρn(G) be the size of a minimum neighborhood cover set and αn(G) of a maximum neighborhood-independent set. Lehel and Tuza defined neighborhood-perfect graphs G as those where the equality ρn(G ) = αn(G ) holds for every induced subgraph G of G.In this work we prove forbidden induced subgraph characterizations of the class of neighborhood-perfect graphs, restricted to two superclasses of cographs: P4-tidy graphs and tree-cographs. We give as well lineartime algorithms for solving the recognition problem of neighborhood-perfect graphs and the problem of finding a minimum neighborhood cover set and a maximum neighborhood-independent set in these same classes.

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Section: Recognition Algorithmsmentioning

confidence: 99%

Section: Further Remarksmentioning

confidence: 99%

Neighborhood covering and independence on $$P_4$$P4-tidy graphs and tree-cographs

2017

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“…It is known that, given a graph of clique-width at most w, an MSO 1 formula ϕ, and an assignment to the free variables in ϕ, the problem of deciding whether G models ϕ with the given assignment is solvable in time O(f (||ϕ||, w) · n 3 ), where f is a computable function and ||ϕ|| is the length of ϕ [3,20]. Proof.…”

Section: Clique-width and The Number Of Movesmentioning

confidence: 99%

How Bad is the Freedom to Flood-It?

Belmonte¹,

Ghadikolaei²,

Kiyomi³

et al. 2019

JGAA

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Fixed-Flood-It and Free-Flood-It are combinatorial problems on graphs that generalize a very popular puzzle called Flood-It. Both problems consist of recoloring moves whose goal is to produce a monochromatic ("flooded") graph as quickly as possible. Their difference is that in Free-Flood-It the player has the additional freedom of choosing the vertex to play in each move. In this paper, we investigate how this freedom affects the complexity of the problem. It turns out that the freedom is bad in some sense. We show that some cases trivially solvable for Fixed-Flood-It become intractable for Free-Flood-It. We also show that some tractable cases for Fixed-Flood-It are still tractable for Free-Flood-It but need considerably more involved arguments. We finally present some combinatorial properties connecting or separating the two problems. In particular, we show that the length of an optimal solution for Fixed-Flood-It is always at most twice that of Free-Flood-It, and this is tight. ACM Subject ClassificationMathematics of computing → Graph algorithms, Theory of computation → Parameterized complexity and exact algorithms

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“…Therefore its restriction to graphs of bounded tree-or cliquewidth can be solved in linear time [8,9].…”

Section: Bounded Cliquewidthmentioning

confidence: 99%

On the Stable Degree of Graphs

Müller

2012

Graph-Theoretic Concepts in Computer Science

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Abstract. We define the stable degree s(G) of a graph G by s(G) = minU maxv∈U dG (v), where the minimum is taken over all maximal independent sets U of G. For this new parameter we prove the following. Deciding whether a graph has stable degree at most k is NP-complete for every fixed k ≥ 3; and the stable degree is hard to approximate. For asteroidal triple-free graphs and graphs of bounded asteroidal number the stable degree can be computed in polynomial time. For graphs in these classes the treewidth is bounded from below and above in terms of the stable degree.

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Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width

Cited by 694 publications

References 35 publications

Neighborhood covering and independence on $$P_4$$P4-tidy graphs and tree-cographs

Neighborhood covering and independence on $$P_4$$P4-tidy graphs and tree-cographs

How Bad is the Freedom to Flood-It?

On the Stable Degree of Graphs

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