Browsing by Author "Sen, J."

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A note on bounds for the broadcast domination number of graphs
(Elsevier B.V., 2024) Sen, J.; Kola, S.R.
A dominating broadcast of a graph G is a function f:V(G)→{0,1,2,…,diam(G)} such that f(v)⩽e(v) for all v∈V(G), where e(v) is the eccentricity of v, and for every u∈V(G), there exists a vertex v with f(v)⩾1 and d(u,v)⩽f(v). The cost of f is ∑v∈V(G)f(v). The minimum cost over all the dominating broadcasts of G is called the broadcast domination number γb(G) of G. In this paper, we give new upper and lower bounds for γb(G). We show that both the bounds are tight. Also, we improve the upper bound for a subclass of regular graphs. Further, we explore the broadcast domination numbers of generalized Petersen graphs and a subclass of circulant graphs. © 2024 Elsevier B.V.
Broadcast Domination in Line Graphs of Trees
(Charles Babbage Research Centre, 2023) Sen, J.; Rao Kola, S.
A dominating broadcast of a graph G is a function f : V(G) → {0, 1, 2, . . ., diam(G)} such that f(v) ≼ e(v) for all v ∈ V(G), where e(v) is the eccentricity of v, and for every vertex u ∈ V(G), there exists a vertex v with f(v) > 0 and d(u, v) ≼ f(v). The cost of f is Pv∈V(G) f(v). The minimum of costs over all the dominating broadcasts of G is called the broadcast domination number γb(G) of G. A graph G is said to be radial if γb(G) = rad(G). In this article, we give tight upper and lower bounds for the broadcast domination number of the line graph L(G) of G, in terms of γb(G), and improve the upper bound of the same for the line graphs of trees. We present a necessary and sufficient condition for radial line graphs of central trees, and exhibit constructions of infinitely many central trees T for which L(T) is radial. We give a characterization for radial line graphs of trees, and show that the line graphs of the i-subdivision graph of K1,n and a subclass of caterpillars are radial. Also, we show that γb(L(C)) = γ(L(C)) for any caterpillar C. © 2023 Charles Babbage Research Centre. All rights reserved.
Broadcast domination of lexicographic and modular products of graphs
(Taylor and Francis Ltd., 2022) Sen, J.; Kola, S.R.
A dominating broadcast labeling of a graph G is a function (Formula presented.) such that (Formula presented.) for all (Formula presented.) where e(v) is the eccentricity of v, and for every vertex (Formula presented.) there exists a vertex v with (Formula presented.) and (Formula presented.) The cost of f is (Formula presented.) The minimum of costs over all the dominating broadcast labelings of G is called the broadcast domination number (Formula presented.) of G. In this paper, we give bounds to the broadcast domination number of lexicographic product G • H of a connected graph G and a graph H, and we show that the bounds are tight by determining the exact values for lexicographic products of some classes of graphs. Also, we give an algorithm which produces a dominating broadcast labeling of G • H. We use the algorithm to find (Formula presented.) • (Formula presented.) and (Formula presented.) • (Formula presented.) Further, we give an upper bound for the broadcast domination number of the modular product of two connected graphs and exact values are determined for the modular product of two graphs involving path, cycle and complete graphs. © 2022 The Author(s). Published with license by Taylor & Francis Group, LLC.
CRITICAL ASPECTS IN BROADCAST DOMINATION
(University of Zielona Gora, 2024) Sen, J.; Kola, S.R.
A dominating broadcast labeling of a graph G is a function f : V (G) → {0, 1, 2, . . . , diam(G)} such that f(v) ≤ e(v) for all v ∈ V (G) and [ v2V (G) f(v)>0 [{u ∈ V (G) : d(u, v) ≤ f(v)}] = V (G), where e(v) is the eccentricity of v. The cost of f is Σ v2V (G) f(v). The minimum of costs over all the dominating broadcast labelings of G is called the broadcast domination number γb(G) of G. In this paper, we introduce the critical aspects in broadcast domination and study it with respect to edge deletion and edge addition. A graph G is said to be k-γ+ b -edge-critical (k-γ- b - edge-critical) if γb(G - e) > γb(G), for every edge e ∈ E(G) (if γb(G + e) < γb(G), for every edge e /∈ E(G)), where γb(G) = k. We give a necessary and sufficient condition for a graph to be k-γ+ b -edge-critical. We characterize k-γ- b -edge-critical graphs for k = 1, 2, and give necessary conditions of the same for k > 3. Further, we define the broadcast bondage number and the broadcast reinforcement number of a graph, and give tight upper bounds for them. © 2024 University of Zielona Gora. All rights reserved.