Bipartization of Graphs

Miotk, Mateusz; Topp, Jerzy; Żyliński, Paweł

doi:10.1007/s00373-019-02068-5

Cited by 2 publications

(2 citation statements)

References 5 publications

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“…Our characterization given in the next theorem is similar but different from those in Hartnell and Rall (1995) and Randerath and Volkmann (1998) [see Theorems 3.6 and 4.1 in Hartnell and Rall (1995), and Theorems 3.3 and 3.4 in Randerath and Volkmann (1998)]. Recently Miotk et al (2019) observed that the graphs belonging to the set B can also be characterized in terms of some graph operations.…”

Section: Bipartite Graphs With the Largest Domination Numbermentioning

confidence: 68%

Graphs with equal domination and covering numbers

et al. 2019

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A dominating set of a graph G is a set D ⊆ V G such that every vertex in V G − D is adjacent to at least one vertex in D, and the domination number γ (G) of G is the minimum cardinality of a dominating set of G. A set C ⊆ V G is a covering set of G if every edge of G has at least one vertex in C. The covering number β(G) of G is the minimum cardinality of a covering set of G. The set of connected graphs G for which γ (G) = β(G) is denoted by C γ =β , whereas B denotes the set of all connected bipartite graphs in which the domination number is equal to the cardinality of the smaller partite set. In this paper, we provide alternative characterizations of graphs belonging to C γ =β and B. Next, we present a quadratic time algorithm for recognizing bipartite graphs belonging to B, and, as a side result, we conclude that the algorithm of Arumugam et al. (Discrete Appl Math 161:1859-1867, 2013) allows to recognize all the graphs belonging to the set C γ =β in quadratic time either. Finally, we consider the related problem of patrolling grids with mobile guards, and show that it can be solved in O(n log n + m) time, where n is the number of line segments of the input grid and m is the number of its intersection points.

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Section: Bipartite Graphs With the Largest Domination Numbermentioning

confidence: 68%

Graphs with equal domination and covering numbers

et al. 2019

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“…Our characterization given in the next theorem is similar but different from those in [12,25] (see Theorems 3.6 and 4.1 in [12], and Theorems 3.3 and 3.4 in [25]). Recently Miotk et al [20] observed that the graphs belonging to the set B can also be characterized in terms of some graph operations.…”

Section: Alternative Characterization Of the Set C γ=βmentioning

confidence: 99%

Graphs with equal domination and covering numbers

Lingas,

Miotk,

Topp

et al. 2018

Preprint

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A dominating set of a graph G is a set D ⊆ V G such that every vertex in V G − D is adjacent to at least one vertex in D, and the domination numberwhile B denotes the set of all connected bipartite graphs in which the domination number is equal to the cardinality of the smaller partite set. In this paper, we provide alternative characterizations of graphs belonging to C γ=β and B. Next, we present a quadratic time algorithm for recognizing bipartite graphs belonging to B, and, as a side result, we conclude that the algorithm of Arumugam et al. [2] allows to recognize all the graphs belonging to the set C γ=β in quadratic time either. Finally, we consider the related problem of patrolling grids with mobile guards, and show that this problem can be solved in O(n log n + m) time, where n is the number of line segments of the input grid and m is the number of its intersection points.

show abstract

Bipartization of Graphs

Cited by 2 publications

References 5 publications

Graphs with equal domination and covering numbers

Graphs with equal domination and covering numbers

Graphs with equal domination and covering numbers

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