Well irredundant graphs

Topp, Jerzy; Vestergaard, Preben Dahl

doi:10.1016/0166-218x(94)00073-9

Cited by 9 publications

(7 citation statements)

References 9 publications

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“…The corona F • K 1 of a graph F is the graph formed from F by adding a new vertex v and edge vv for each vertex v of F. A graph G is said to be a corona graph if G = F • K 1 for some graph F. We note that a graph G is a corona graph if and only if each vertex of G is a leaf or it is adjacent to exactly one leaf of G, and consequently every corona graph belongs to the set C γ =β . Payan and Xuong (1982) and Fink et al (1985) have proved that for a connected graph G of even order is γ (G) = |V G |/2 if and only if G is the cycle C 4 or the corona F • K 1 for any connected graph F [see also Topp and Vestergaard (1995) for a short proof]. From this (or directly from Theorem 2) we immediately have the following corollary.…”

Section: Corollarymentioning

confidence: 82%

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: Corollarymentioning

confidence: 82%

Graphs with equal domination and covering numbers

et al. 2019

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“…We note that a graph G is a corona graph if and only if each vertex of G is a leaf or it is adjacent to exactly one leaf of G, and consequently every corona graph belongs to the set C γ=β . Payan and Xuong [22], and Fink et al [10] have proved that for a connected graph G of even order is γ(G) = |V G |/2 if and only if G is the cycle C 4 or the corona F • K 1 for any connected graph F (see also [29] for a short proof). From this (or directly from Theorem 5) we immediately have the following corollary.…”

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.

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“…The corresponding polynomial algorithms follow directly from the above definitions. Other classes of graphs having a polynomial time algorithm for calculating the domination and independent domination numbers can be found in [1,2,3,4,5,6,8,12,14,16,17].…”

Section: Theorem D Any Upper Domination Perfect Graph Is Upper Irredumentioning

confidence: 99%

Locally Well-Dominated and Locally Independent Well-Dominated Graphs

Zverovich

2003

Graphs and Combinatorics

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In this article we present characterizations of locally well-dominated graphs and locally independent well-dominated graphs, and a sufficient condition for a graph to be k-locally independent well-dominated. Using these results we show that the irredundance number, the domination number and the independent domination number can be computed in polynomial time within several classes of graphs, e.g., the class of locally well-dominated graphs.

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Well irredundant graphs

Cited by 9 publications

References 9 publications

Graphs with equal domination and covering numbers

Graphs with equal domination and covering numbers

Graphs with equal domination and covering numbers

Locally Well-Dominated and Locally Independent Well-Dominated Graphs

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