DOMINATING SET remains NP-complete even when instances are restricted to bipartite graphs, however, in this case VERTEX COVER is solvable in polynomial time. Consequences to VECTOR DOMINATING SET as a generalization of both are discussed

Gerlach, Tobias

Harant, Jochen

Discussiones Mathematicae Graph Theory

English

Digitale Bibliothek Thüringen

   TU Ilmenau | Universitätsbibliothek | ilmedia, 2020 http://www.tu-ilmenau.de/ilmedia Gerlach, Tobias; Harant, Jochen: A note on domination in bipartite graphs URN: urn:nbn:de:gbv:ilm1-2020200083  Original published in: Discussiones mathematicae. Graph theory / Uniwersytet Zielonogórski, Wydział Matematyki, Informatyki i Ekonometrii. - Warsaw : De Gruyter Open. - 22 (2002), 2, p. 229-231. Original published: 2002 ISSN: 2083-5892 DOI: 10.7151/dmgt.1171 [Visited: 2020-01-17]     This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported license.  To view a copy of this license, visit  http://creativecommons.org/licenses/BY-NC-ND/3.0/ Discussiones Mathematicae 229Graph Theory 22 (2002 ) 229–231A NOTE ON DOMINATION IN BIPARTITE GRAPHSTobias Gerlach and Jochen HarantDepartment of MathematicsTechnical University of IlmenauD-98684 Ilmenau, GermanyAbstractDOMINATING SET remains NP -complete even when instancesare restricted to bipartite graphs, however, in this case VERTEXCOVER is solvable in polynomial time. Consequences to VECTORDOMINATING SET as a generalization of both are discussed.Keywords: bipartite graph, domination.2000 Mathematics Subject Classification: 05C35.For terminology and notation not defined here we refer to [2, 3]. Givena finite, simple, and undirected graph G without isolated vertices, V (G) ={1, . . . , n}, E(G), G[V ], and di(G) denote its vertex set, its edge set, the sub-graph of G induced by V ⊆ V (G), and the degree of i ∈ V (G) in G, respec-tively. Furthermore, let ~1 = (1, . . . , 1) ∈ Rn and ~d(G) = (d1(G), . . . , dn(G)).Given an integral vector ~k = (k1, . . . , kn) with 1 ≤ ki ≤ di(G) for i =1, . . . , n, a set D ⊆ V (G) is called ~k-dominating if di(G[V (G) \ D]) ≤di(G) − ki for i ∈ V (G) \ D. Consider the corresponding decision prob-lem, which was investigated in [4]:VECTOR DOMINATING SETInstance: A graph G on V (G) = {1, . . . , n}, an integral vector ~k =(k1, . . . , kn) with 1 ≤ ki ≤ di(G) for i = 1, . . . , n, and a positive integer l.Question: Does G contain a ~k-dominating set D with |D| ≤ l?The restriction of VECTOR DOMINATING SET to ~k = ~1 is the decisionproblem DOMINATING SET, remaining NP -complete even when instancesare restricted to bipartite graphs [1]. It is easy to see that D ⊆ V (G) is ~d(G)-dominating if and only if each edge of G has at least one endvertex in D.230 T. Gerlach and J. HarantThe corresponding decision problem VERTEX COVER is NP -complete ingeneral, however, is solvable in polynomial time for bipartite graphs [3].The question arises in how many components ~k may differ from ~1 or from~d(G) such that the restriction of VECTOR DOMINATING SET to bipartitegraphs remains NP -complete or is solvable in polynomial time, respectively.Theorem 1 and Theorem 2 give partial answers to this question.Theorem 1. Given 0 < c < 12 , the following restriction of VECTORDOMINATING SET remains NP -complete.Instance: A bipartite graph G on V (G) = {1, . . . , n}, an integral vector ~k =(k1, . . . , kn) with 1 ≤ ki ≤ di(G) for i = 1, . . . , n and |{i ∈ V (G) | ki > 1}|= dcne, and a positive integer l.Question: Does G contain a ~k-dominating set D with |D| ≤ l?Theorem 2. Given c > 0, the following restriction of VECTOR DOMI-NATING SET is solvable in polynomial time.Instance: A bipartite graph G on V (G) = {1, . . . , n}, an integral vector ~k =(k1, . . . , kn) with 1 ≤ ki ≤ di(G) for i = 1, . . . , n and |E(G[{i ∈ V (G) | ki <di(G)}])| ≤ c log2n, and a positive integer l.Question: Does G contain a ~k-dominating set D with |D| ≤ l?Proof of Theorem 1. Given 0 < c < 12 , we shall transform DOMI-NATING SET to the restriction of VERTEX DOMINATING SET ofTheorem 1. Let a graph H on V (H) = {1, . . . ,m} and a positive integer lbe an instance of DOMINATING SET, the positive integer r be choosensuch that 2cm1−2c ≤ r < 2cm1−2c + 1, hence, c(2m + 2r) ≤ r < c(2m + 2r) +(1 − 2c) implying r = dc(2m + 2r)e ≥ 1, and G be constructed withV (G) = {1, . . . , 2m+2r} and E(G) = {(i,m+ j), (j,m+ i) | (i, j) ∈ E(H)}∪{(i,m+i), (m+i, 2m+j), (2m+j, 2m+r+j) | i = 1, . . . ,m, j = 1, . . . , r}.Following the ideas in [1], it is easy to see that G is bipartite and that forD(H) ⊆ V (H), D(G) = {i + m | i ∈ D(H)} ∪ {2m + 1, . . . , 2m + r} is a~1-dominating set of G if and only if D(H) is a ~1-dominating set of H. Let~k = (k1, . . . , k2m+2r) be an arbitrary integral vector with 1 ≤ kp ≤ dp(G) forp = 1, . . . , 2m+2r, and kp = 1 if and only if p ∈ V (G)\{2m+1, . . . , 2m+r}.Then |{i ∈ V (G) | ki > 1}| = r = dc|V (G)|e, and D(G) is even a ~k-domi-nating set of G if D(H) is a ~1-dominating set of H. Hence, H contains a~1-dominating set of cardinality l if and only if G contains a ~k-dominating setof cardinality l+r. With |V (G)| = 2m+2r < 2m+2c(2m+2r)+2(1−2c) =2|V (H)|+2c|V (G)|+2(1−2c), hence, |V (G)| < 21−2c |V (H)|+2 we are done.Note on Domination in Bipartite Graphs 231Proof of Theorem 2. For D ⊆ V (G), let E(D) be the set of edgeshaving no endvertex in D, andH(D) be the graph arising from G by deletingthe edges of E(D). It is easy to see that D is a ~k-dominating set of G if andonly if E(D) ⊆ E(G[{i ∈ V (G) | ki < di(G)}]), each endvertex i of an edge inE(D) is endvertex of at most di(G)−ki edges in E(D), and D is a ~d(H(D))-dominating set of H(D). Since VERTEX COVER is solvable in polynomialtime and the number of sets E(D) is at most 2|E(G[{i∈V (G) | ki<di(G)}])| ≤ ncwe are done.Remark. It remains open whether the bounds cn (0 < c < 12) and c log2n(c > 0) of Theorem 1 and Theorem 2 can be made significantly greatersuch that VECTOR DOMINATING SET is still NP-complete or solvable inpolynomial time, respectively.References[1] G.J. Chang and G.L. Nemhauser, The k-domination and k-stability problemsin sun-free chordal graphs, SIAM J. Algebraic Discrete Methods 5 (1984)332–345.[2] R. Diestel, Graph Theory (Springer-Verlag, New York, 2000).[3] M.R. Garey and D.S. Johnson, Computers and Intractability (W.H. Freemanand Company, San Francisco, 1979).[4] J. Harant, A. Pruchnewski and M. Voigt, On dominating sets and independentsets of graphs, Combinatorics, Probability and Computing 8 (1999) 547–553.Received 24 August 2000Powered by TCPDF (www.tcpdf.org)

A   note on domination in bipartite graphs

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A note on domination in bipartite graphs

Tobias Gerlach

Jochen. Harant

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A note on domination in bipartite graphs

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A note on domination in bipartite graphs

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