Safe Sets in Graphs: Graph Classes and Structural Parameters

Águeda, Raquel; Cohen, Nathann; Fujita, Shinya; Legay, Sylvain; Manoussakis, Yannis; Matsui, Yasuko; Montero, Leandro; Naserasr, Reza; Otachi, Yota; Sakuma, Tadashi; Tuza, Źsolt; Xu, Renyu

doi:10.1007/978-3-319-48749-6_18

Cited by 5 publications

(7 citation statements)

References 11 publications

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“…It would be more preferable if one can find a polynomial‐time approximation scheme (PTAS) computing a (connected) safe set with minimum weight of a weighted graph of bounded treewidth. Although a pseudo‐polynomial‐time algorithm is easily obtained from the polynomial‐time algorithm for the cardinality version in , it seems to be difficult to modify this pseudo‐polynomial‐time algorithm to be a PTAS. It would be worth characterizing the class of graphs

G = (V, E)

such that

cs (G, w) = s (G, w)

holds for any positive weight function w on V . As mentioned in Lemma 1, the stars

T = (V, E)

belong to this class.…”

Section: Resultsmentioning

confidence: 99%

“…As is proven in , both the problem of computing the safe number and the problem of computing the connected safe number are

N P

‐hard in general while the connected safe number of a tree can be computed in linear time. Quite recently, by using dynamic programming , the authors in obtained an

O (n^{5})

‐time algorithm for finding a safe set with minimum cardinality of a tree with n vertices. By using the same method, they also proved that both the safe number

s (G)

and the connected safe number

cs (G)

of a given graph G of bounded treewidth can be computed in polynomial‐time.…”

Section: Introductionmentioning

confidence: 99%

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Safe sets, network majority on weighted trees

et al. 2017

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Let G = (V , E) be a graph and let w : V → R >0 be a positive weight function on the vertices of G. induced by a weighted safe set S is connected, then the set S is called a weighted connected safe set. In this article, we show that the problem of computing the minimum weight of a safe set is N P-hard for trees, even if the underlying tree is restricted to be a star, but it is polynomially solvable for paths. We also give an O(n log n) time 2-approximation algorithm for finding a weighted connected safe set with minimum weight in a weighted tree. Then, as a generalization of the concept of a minimum safe set, we define the concept of a parameterized infinite family of proper central subgraphs on weighted trees, whose polar ends are the vertex set of the tree and the centroid points. We show that each of these central subgraphs includes a centroid point. K E Y W O R D Sapproximation algorithm, centroid, network majority, N P-completeness, safe set, weighted tree

show abstract

G = (V, E)

such that

cs (G, w) = s (G, w)

holds for any positive weight function w on V . As mentioned in Lemma 1, the stars

T = (V, E)

belong to this class.…”

Section: Resultsmentioning

confidence: 99%

“…As is proven in , both the problem of computing the safe number and the problem of computing the connected safe number are

N P

‐hard in general while the connected safe number of a tree can be computed in linear time. Quite recently, by using dynamic programming , the authors in obtained an

O (n^{5})

‐time algorithm for finding a safe set with minimum cardinality of a tree with n vertices. By using the same method, they also proved that both the safe number

s (G)

and the connected safe number

cs (G)

of a given graph G of bounded treewidth can be computed in polynomial‐time.…”

Section: Introductionmentioning

confidence: 99%

Safe sets, network majority on weighted trees

et al. 2017

Self Cite

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“…But, on the contrary, Fujita et al [2] show that cs(G) can be computed in linear time in case of trees. Also, Árueda et al [4] show that s(G) can be computed in O(n 5 ) time on trees. Any tree T with one vertex of degree not more than 3 holds that s(T) � cs(T).…”

Section: Introductionmentioning

confidence: 99%

On the Connected Safe Number of Some Classes of Graphs

Iqbal

Sardar

Alrowaili

et al. 2021

Journal of Mathematics

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For a connected simple graph G , a nonempty subset S of V G is a connected safe set if the induced subgraph G S is connected and the inequality S ≥ D satisfies for each connected component D of G∖S whenever an edge of G exists between S and D . A connected safe set of a connected graph G with minimum cardinality is called the minimum connected safe set and that minimum cardinality is called the connected safe numbers. We study connected safe sets with minimal cardinality of the ladder, sunlet, and wheel graphs.

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“…Moreover mainly theoretical contributions on the subject can be found. For example, the weighted problem over trees has been shown to be NP-hard in [7,8] while the unweighted problem is shown to be solvable in polynomial time on graphs with bounded treewidth and interval graphs in [1]. Bounds on the size of safe sets and connected safe sets are also studied in [13].…”

Section: Introductionmentioning

confidence: 99%

A compact mixed integer linear formulation for safe set problems

Hosteins¹

2020

Optim Lett

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The Safe Set Problem is a type of partitioning problem with particular constraints on the weight of adjacent partition components. Several theoretical results exist in the literature along with a mixed integer linear formulation. We propose a new, more compact, Mixed Integer Linear Program for the (Weighted) Safe Set Problem and Connected Safe Set Problem with a polynomial number of variables and constraints, based on a flow from an additional fictitious node. The model is enriched by symmetrybreaking considerations and a variable reduction subproblem. We test the model on a benchmark of small instances which underline the difficulty of solving the Safe Set Problem through mathematical programming approaches. We also compare it with the only existing formulation in the literature and find that our approach is usually more efficient on a set of benchmark instances.

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Safe Sets in Graphs: Graph Classes and Structural Parameters

Cited by 5 publications

References 11 publications

Safe sets, network majority on weighted trees

Safe sets, network majority on weighted trees

On the Connected Safe Number of Some Classes of Graphs

A compact mixed integer linear formulation for safe set problems

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