Abstract
An outer-connected dominating set in a graph G = (V, E) is a set of vertices D ⊆ V satisfying the condition that, for each vertex v ∉ D, vertex v is adjacent to some vertex in D and the subgraph induced by V∖D is connected. The outer-connected dominating set problem is to find an outer-connected dominating set with the minimum number of vertices which is denoted by \(\tilde {\gamma }_{c}(G)\). In this paper, we determine \(\tilde {\gamma }_{c}(S(n,k))\), \(\tilde {\gamma }_{c}(S^{+}(n,k))\), \(\tilde {\gamma }_{c}(S^{++}(n,k))\), and \(\tilde {\gamma }_{c}(S_{n})\), where S(n, k), S +(n, k), S ++(n, k), and S n are Sierpi\(\acute {\mathrm {n}}\)ski-like graphs.
Similar content being viewed by others
References
Akhbari, M.H., Hasni, R., Favaron, O., Karami, H., Sheikholeslami, S.M.: On the outer-connected domination in graphs. J. Comb. Optim. 26, 10–18 (2013)
Chen, G.H., Duh, D.R.: Topological properties, communication, and computation on WK-recursive networks. Networks 24, 303–317 (1994)
Cyman, J.: The outer-connected domination number of a graph. Australas. J. Comb. 38, 35–46 (2007)
Duh, D.R., Chen, G.H.: Topological properties of WK-recursive networks. J. Parallel Distrib. Comput. 23, 468–474 (1994)
Hinz, A.M., Schief, A.: The average distance on the Sierpiński gasket. Probab. Theory Relat. Fields 87, 129–138 (1990)
Hinz, A.M.: Pascal’s triangle and the Tower of Hanoi. Am. Math. Mon. 99, 538–544 (1992)
Hinz, A.M., Klavžar, S., Milutinoviá, U., Parisse, D., Petr, C.: Metric properties of the Tower of Hanoi graphs and Stern’s diatomic sequence. Eur. J. Comb. 26, 693–708 (2005)
Hinz, A.M., Parisse, D.: The Average Eccentricity of Sierpiński Graphs. Graphs and Combinatorics 28, 671–686 (2012)
Hinz, A.M., Klavžar, S., Milutinović, U., Petr, C.: The Tower of Hanoi-Myths and Maths. Birkhäuser/Springer Basel AG, Basel (2013)
Jakovac, M., Klavžar, S.: Vertex-, edge-, and total-colorings of Sierpiński-like graphs. Discret. Math. 309, 1548–1556 (2009)
Jiang, H., Shan, E.: Outer-connected domination number in graphs. Utilitas Math. 81, 265–274 (2010)
Kaimanovich, V.A. In: Grabner, P., W. Woess (eds.): Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization, In: Fractals in Graz 2001, pp 145–183. Birkhaüser (2003)
Klavžar, S., Milutinović, U.: Graphs S(n, k) and a variant of the Tower of Hanoi problem. Czechoslov. Math. J. 47, 95–104 (1997)
Klavžar, S., Milutinović, U., Petr, C.: 1-perfect codes in Sierpiński graphs. Bull. Aust. Math. Soc. 66, 369–384 (2002)
Klavžar, S., Mohar, B.: Crossing numbers of Sierpiński-like graphs. J.Graph Theory 50, 186–198 (2005)
Klavžar, S.: Coloring Sierpiński graphs and Sierpiński gasket graphs. Taiwan. J. Math. 12, 513–522 (2008)
Klix, F., Rautenstrauch-Goede, K.: Struktur-und Komponentenanalyse von Problemlösungsprozessen, Zeitschrift für Psychologie 174, 167–193 (1967)
Lin, C.H., Liu, J.J., Wang, Y.L., Yen, W.C.K.: The hub number of Sierpiński-like graphs. Theory Comput. Syst. 49(3), 588–600 (2011)
Lin, C.H., Liu, J.J., Wang, Y.L.: Global strong defensive alliances of Sierpiński-like graphs. Theory Comput. Syst. 53(3), 365–385 (2013)
MarkKeil, J., Pradhan, D.: Computing a minimum outer-connected dominating set for the class of chordal graphs. Information Processing Letters 113, 552–561 (2013)
Parisse, D.: On some metric properties of the Sierpiński graphs S(n, k). Ars Combinatoria 90, 145–160 (2009)
Romik, D.: Shortest paths in the Tower of Hanoi graph and finite automata. SIAM J. Discret. Math. 20, 610–622 (2006)
Sydow, H.: Zur metrischen Erfasung von subjektiven Problemzuständen und zu deren Veränderung im Denkprozes, Zeitschrift für Psychologie 177, 145–198 (1970)
Teguia, A.M., Godbole, A.P.: Sierpiński gasket graphs and some of their properties. Australas. J. Comb. 35, 181–192 (2006)
Vecchia, G.D., Sanges, C.: A recursively scalable network VLSI implementation. Futur. Gener. Comput. Syst. 4, 235–243 (1988)
Acknowledgments
The authors would like to thank anonymous referees for their careful reading with corrections and useful comments which helped to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported in part by the National Science Council of Republic of China under contracts NSC 100–2221–E–011–067-MY3, NSC 101–2221–E–011–038–MY3, NSC 100-2221-E-011-068-, and NSC 100-2221-E-128-003-.
Rights and permissions
About this article
Cite this article
Chang, SC., Liu, JJ. & Wang, YL. The Outer-connected Domination Number of Sierpiński-like Graphs. Theory Comput Syst 58, 345–356 (2016). https://doi.org/10.1007/s00224-015-9621-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-015-9621-9