Acyclic domination on bipartite permutation graphs

Xu, Guangjun; Kang, Liying

doi:10.1016/j.ipl.2006.04.002

Cited by 6 publications

(4 citation statements)

References 8 publications

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“…In a bipartite graph G = (X ∪ Y, E), a strong ordering of the vertices of G consists of an ordering of X and an ordering of Y such that for all (x, y ) and (x , y) in E, where x, x ∈ X and y, y ∈ Y , if x < x and y < y , then (x, y) and (x , y ) are in E. A bipartite graph G = (X ∪ Y, E) is a bipartite permutation graph if there exists a strong ordering of X ∪ Y [7,10].…”

Section: Lemma 3 G R Is the Line Graph Of G Bmentioning

confidence: 99%

The rook problem on saw-toothed chessboards

Chen

2008

Applied Mathematics Letters

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A saw-toothed chessboard, or STC for short, is a kind of chessboard whose boundary forms two staircases from left down to right without any hole inside it. A rook at square (i, j) can dominate the squares in row i and in column j. The rook problem of an STC is to determine the minimum number of rooks that can dominate all squares of the STC. In this paper, we model an STC by two particular graphs: a rook graph and a board graph. We show that for an STC, the rook graph is the line graph of the board graph, and the board graph is a bipartite permutation graph. Thus, the rook problem on STCs can be solved by any algorithm for solving the edge domination problem on bipartite permutation graphs.

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Section: Lemma 3 G R Is the Line Graph Of G Bmentioning

confidence: 99%

The rook problem on saw-toothed chessboards

Chen

2008

Applied Mathematics Letters

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“…Similarly, requiring D to have no edges means that D is an independent dominating set. And there are more general ideas such as acyclic domination, introduced by Hedetniemi, Hedetniemi and Rall [3] and explored further for example in [8].…”

Section: Introductionmentioning

confidence: 99%

Acyclic total dominating sets in cubic graphs

Goddard¹,

Henning²

2019

APPL ANAL DISCRETE M

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“…In particular, they proved that Acyclic Dominating Set can be solved in polynomial time in interval graphs and proper circular-arc graphs. Xu, Kang, and Shan [34] proved that Acyclic Dominating Set is linear-time solvable in bipartite permutation graphs. The complexity status of Acyclic Dominating Set in circle graphs was unknown.…”

Section: Introductionmentioning

confidence: 99%

Parameterized Domination in Circle Graphs

et al. 2013

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A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Appl. Math., 42(1):51-63, 1993] proved that DOMINATING SET, CONNECTED DOMINATING SET, and TOTAL DOMINATING SET are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the param-eterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: A preliminary conference version of this work appeared in the Theory Comput Syst • DOMINATING SET, INDEPENDENT DOMINATING SET, CONNECTED DOMINATING SET, TOTAL DOMINATING SET, and ACYCLIC DOMINATING SET are W [1]-hard in circle graphs, parameterized by the size of the solution. • Whereas both CONNECTED DOMINATING SET and ACYCLIC DOMINATING SET are W [1]-hard in circle graphs, it turns out that CONNECTED ACYCLIC DOMINATING SET is polynomial-time solvable in circle graphs. • If T is a given tree, deciding whether a circle graph G has a dominating set inducing a graph isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by t = |V (T)|. We prove that the FPT algorithm runs in subexpo-nential time, namely 2 O(t· log log t log t) · n O(1) , where n = |V (G)|.

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Acyclic domination on bipartite permutation graphs

Cited by 6 publications

References 8 publications

The rook problem on saw-toothed chessboards

The rook problem on saw-toothed chessboards

Acyclic total dominating sets in cubic graphs

Parameterized Domination in Circle Graphs

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