Figure 6.3: An illustration of the signed star domination number of Cay(Z15,{3, 5, 10, 12}) and the corre-sponding SSDF.
6.4 Signed star domination of
{2, 1}-factorable graphs
Before going further, we introduce a new definition: A graph is {2, 1}-factorable if it can be decomposed into 2-factors and/or 1-factors. For the definitions of factor”and “k-factorable”refer to [79].
Theorem 6.4.1. All Cayley graphs are {2, 1}-factorable graphs.
Proof. For Cayley graphs G = Cay(Γ, Ω),Ci’s andMj’s (defined in Section 6.3) are 2-factors and 1-factors, respectively, and hence G is {2, 1}-factorable.
Notice that although every Cayley graph is {2, 1}-factorable, a {2, 1}-factorable graph may not be a Cayley graph. For example, the Petersen graph and the dodecahedral graph are {2, 1}-factorable but not Cayley graphs [60]. Notice also that a {2, 1}-factorable graph
CHAPTER 6. THE SIGNED STAR DOMINATION 6.4. SSD OF {2, 1}-FACTORABLE GRAPHS
is regular but it is not necessarily 1-factorable; the Petersen graph is an example.
The property of Cayley graphs used in the proofs of Theorems 6.3.4, 6.3.5 and 6.3.6 is that: A Cayley graph can be decomposed into a bunch of Ci’s and Mj’s. Since Ci’s are 2-factors and Mj’s are 1-factors, we now extend Theorems 6.3.4, 6.3.5 and 6.3.6 to {2, 1}-factorable graphs in the following three theorems. We will omit the proofs of these theorems since we can replace Ci’s with 2-factors and Mj’s with 1-factors in the proofs of Theorems 6.3.4, 6.3.5 and 6.3.6.
Theorem 6.4.2. Let G be an r-regular and {2, 1}-factorable graph of order n. If r is odd, then γSS(G) = n2.
Theorem 6.4.3. Let G be an r-regular and {2, 1}-factorable graph of order n with even r.
If there exists a{2, 1}-decomposition of G containing odd number of 2-factors, or a 1-factor, or a 2-factor consisting of even cycles, then γSS(G) = n.
Theorem 6.4.4. Let G be an r-regular and 2-factorable graph of odd order n. If there exists a 2-factorization of G containing a Hamiltonian cycle and r is a multiple of 4, then γSS(G) = n + 1.
We now have a conjecture.
Conjecture 6.4.5. Let G be an r-regular and 2-factorable graph of order n. Then
γSS(G) =
n if r≡ 2 (mod 4) or n is even, n + 1 if r≡ 0 (mod 4) and n is odd.
Chapter 7 Conclusions
In this chapter, we present a summary of this thesis, and we discuss some directions for further research.
A distributed system is self-stabilizing if, regardless of the initial state, the system guaran-tees to reach a legitimate state in a finite time. In 2007, Turau proposed the first linear-time self-stabilizing algorithm for the minimal dominating set (MDS) problem under an unfair distributed daemon [75]; this algorithm stabilizes in at most 9n moves, where n is the num-ber of nodes. In 2008, Goddard et al. [30] proposed a 5n-move algorithm. The main result of this thesis is a (4n− 2)-move self-stabilizing algorithm for the MDS problem using an unfair distributed daemon; the bound 4n− 2 is tight. The model that we use is the normal model, also called the distance-1 model. It is challenging to design a self-stabilizing MDS using a distributed daemon that makes fewer than 4n− 2 moves.
While MIS algorithms can only stabilize with an independent set, MDS algorithms are capable of being stable with any minimal dominating set. In [36], the authors mentioned that the significance of MDS algorithms is that if the system is initialized to any minimal dominating set with the correct variable settings (including minimal dominating sets that are not independent), then it will remain stable. They also pointed out that an MDS
67
CHAPTER 7. CONCLUSIONS
algorithm is usually more complex than an MIS algorithm but can potentially produce any minimal dominating set. Following this idea, we give four different levels of stableness of self-stabilizing algorithms. We also prove that if an MDS-silent algorithm is preferred, then distance-1 knowledge is insufficient, where a self-stabilizing MDS algorithm is MDS-silent if it will not make any move when the starting configuration of the system is already an MDS. We conjecture that if we relax the MDS-silent property to MDS-stable (the execution of non-membership moves is allowed), then there will not exist an MDS-stable algorithm in the normal model.
We also consider developing self-stabilizing MDS-silent algorithms. In [21] and [76], the authors considered the distance-2 model, in which every node can read the states of nodes up to distance 2; see also [31] for the distance-k model. In particular, [21] proposed an n(k + 1)-move self-stabilizing minimal {k}-dominating set algorithm; when k = 1, the algorithm finds an MDS using at most 2n moves. The paper [76] presented a 2n-move self-stabilizing minimal k-dominating set algorithm; when k = 1, this algorithm also finds an MDS. However, the algorithms in both [21] and [76] operate correctly only with a central daemon. In this thesis, we present an algorithm, which is also 2n-move but under an unfair distributed daemon and hence is more practical. It is easy to generalize our MDS-silent algorithm to self-stabilizing minimal {k}-dominating or k-dominating set algorithms under an unfair distributed daemon for k≥ 2.
Notice that the transformed version of algorithm A1 of [76] is not MDS-stable, even the algorithm is transformed by the transformerC (here we set k = 1). For an counterexample, consider a graph with five nodes v1, v2, v3, v4, and v5 lying on a path in order. Let the states of v1, v2, . . . , v5 be OUT-IN-IN-OUT-IN. If vi.IN count are correct for all i, then only rule R2 of v3 is enabled. By the transformer C, all nodes are consistent and v3 makes a request. After that v2 and v4 approve it. Suppose then v5 fails (no longer alive) just before v3 executes the leaving rule R2. At the mean time, v4 wants to make an update urgently,
but unfortunately the central daemon of the distance-1 model chooses v3 to make a move.
Since the transformed algorithm AC1 makes a membership move, AC1 does not satisfy the MDS-stable property. However, the expression model assumes distance-2 knowledge and therefore v3 can see the failure of v5 and v3 will not leave the MDS; thus the transformer may not derive the same result as the one in the distance-2 model.
Chapter 6 has provided the first study of the problem of finding the values of signed star domination number of Cayley digraphs and Cayley graphs. We define the directed version of an SSDF on a digraph D and give the value of signed star domination number of D. We also obtain exact values for the signed star domination number for certain classes of Cayley graphs. Using these results on Cayley graph, we deduce the signed star domination number of Cn, Kn, and Kn,n. We also generalize the notion of k-factorable graphs to{2, 1}-factorable graphs, in which we address the signed star domination number. However, there remains some Cayley graphs whose signed star domination number are unknown. Another future work is to address the signed star domatic number for Cayley graphs.
69
References
[1] Khaled M. Alzoubi, Peng-Jun Wan, and Ophir Frieder. Maximal independent set, weakly-connected dominating set, and induced spanners in wireless ad hoc networks.
International Journal of Foundations of Computer Science, 14(2):287–303, 2003.
[2] Gheorghe Antonoiu and Pradip K. Srimani. Distributed self-stabilizing algorithm for minimum spanning tree construction. In European Conference on Parallel Processing, pages 480–487. Springer-Verlag, 1997.
[3] M. Atapour, S. M. Sheikholeslami, A.N. Ghameshlou, and L. Volkmann. Signed star domatic number of a graph. Discrete Applied Mathematics, 158(3):213–218, 2010.
[4] Yacine Belhoul, Said Yahiaoui, and Hamamache Kheddouci. Efficient self-stabilizing al-gorithms for minimal total k-dominating sets in graphs. Information Processing Letters, 114(7):339–343, June 2014.
[5] Steven C. Bruell, Sukumar Ghosh, Mehmet Hakan Karaata, and Sriram V. Pemmaraju.
Self-stabilizing algorithms for finding centers and medians of trees. SIAM Journal on Computing, 29(2):600–614, October 1999.
[6] Franck Butelle, Christian Lavault, and Marc Bui. A uniform self-stabilizing minimum diameter spanning tree algorithm. In Jean-Michel H´elary and Michel Raynal, editors,
Distributed Algorithms, volume 972 of Lecture Notes in Computer Science, pages 257–
272. Springer Berlin Heidelberg, 1995.
[7] S´ebastien Cantarell, AjoyK. Datta, and Franck Petit. Self-stabilizing atomicity refine-ment allowing neighborhood concurrency. In Self-Stabilizing Systems, volume 2704 of Lecture Notes in Computer Science, pages 102–112. Springer Berlin Heidelberg, 2003.
[8] Subhendu Chattopadhyay, Lisa Higham, and Karen Seyffarth. Dynamic and self-stabilizing distributed matching. In Proceedings of the 21st Annual Symposium on Principles of Distributed Computing, PODC’02, pages 290–297, New York, NY, USA, 2002. ACM.
[9] Thirugnanam Tamizh Chelvam, G. Kalaimurugan, and Well Y. Chou. The sighed star domination number of Cayley graphs. Discrete Mathematics, Algorithms and Applica-tions, 04(02):1250017, 2012.
[10] Nian-Shing Chen, Hwey-Pyng Yu, and Shing-Tsaan Huang. A self-stabilizing algorithm for constructing spanning trees. Information Processing Letters, 39(3):147–151, 1991.
[11] Well Y. Chiu and Chiuyuan Chen. Linear-time self-stabilizing algorithms for minimal domination in graphs. In Combinatorial Algorithms, volume 8288 of Lecture Notes in Computer Science, pages 115–126. Springer Berlin Heidelberg, 2013.
[12] Well Y. Chiu, Chiuyuan Chen, and Shih-Yu Tsai. A 4n-move self-stabilizing algorithm for the minimal dominating set problem using an unfair distributed daemon. Informa-tion Processing Letters, 114(10):515–518, 2014.
[13] Ajoy K. Datta, Lawrence L. Larmore, and Priyanka Vemula. A self-stabilizing O(k)-time k-clustering algorithm. The Computer Journal, 53(3):342–350, March 2010.
71
[14] Lyes Dekar and Hamamache Kheddouci. Distance-2 self-stabilizing algorithm for a b-coloring of graphs. In Sandeep Kulkarni and Andr´e Schiper, editors, Stabilization, Safety, and Security of Distributed Systems, volume 5340 of Lecture Notes in Computer Science, pages 19–31. Springer Berlin / Heidelberg, 2008.
[15] Edsger W. Dijkstra. Self-stabilizing systems in spite of distributed control. Communi-cations of the ACM, 17(11):643–644, November 1974.
[16] Shlomi Dolev. Self-stabilization. MIT Press, 2000.
[17] Shlomi Dolev and Ted Herman. Superstabilizing protocols for dynamic distributed systems. In Proceedings of the 2nd Workshop on Self-Stabilizing Systems, WSS’95, pages 3.1–3.15, 1995.
[18] Shlomi Dolev, Amos Israeli, and Shlomo Moran. Self-stabilization of dynamic systems assuming only read/write atomicity. Distributed Computing, 7(1):3–16, November 1993.
[19] Shlomi Dolev, Amos Israeli, and Shlomo Moran. Resource bounds for self-stabilizing message-driven protocols. SIAM Journal on Computing, 26(1):273–290, February 1997.
[20] Shlomi Dolev and Nir Tzachar. Empire of colonies: Self-stabilizing and self-organizing distributed algorithm. Theoretical Computer Science, 410(6–7):514–532, February 2009.
[21] Martin Gairing, Wayne Goddard, Stephen T. Hedetniemi, Petter Kristiansen, and Al-ice A. McRae. Distance-two information in self-stabilizing algorithms. Parallel Process-ing Letters, 14(3–4):387–398, 2004.
[22] Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, NY, USA, 1979.
[23] Felix C. Gartner. A survey of self-stabilizing spanning-tree construction algorithms.
Technical report, Swiss Federal Institute of Technology Tech. Rep. IC/2003/38, School of Computer and Communication Sciences, Lausanne, Switzerland, 2003.
[24] Cyril Gavoille, Ralf Klasing, Adrian Kosowski, Lukasz Kuszner, and Alfredo Navarra.
On the complexity of distributed graph coloring with local minimality constraints. Net-works, 54(1):12–19, August 2009.
[25] Sukumar Ghosh, Arobinda Gupta, Ted Herman, and Sriram V. Pemmaraju. Fault-containing self-stabilizing algorithms. In Proceedings of the 15th Annual ACM Sympo-sium on Principles of Distributed Computing, PODC’96, pages 45–54, New York, NY, USA, 1996. ACM.
[26] Sukumar Ghosh and Mehmet Hakan Karaata. A self-stabilizing algorithm for coloring planar graphs. Distributed Computing, 7(1):55–59, November 1993.
[27] Wayne Goddard, Stephen T. Hedetniemi, David P. Jacobs, and Pradip K. Srimani.
A robust distributed generalized matching protocol that stabilizes in linear time. In Proceedings of the 23rd International Conference on Distributed Computing Systems, ICDCSW’03, pages 461–465, Washington, DC, USA, 2003. IEEE Computer Society.
[28] Wayne Goddard, Stephen T. Hedetniemi, David P. Jacobs, and Pradip K. Srimani.
A self-stabilizing distributed algorithm for minimal total domination in an arbitrary system graph. In Proceedings of the 8th International Symposium on Parallel and Dis-tributed Processing, IPDPS’03, pages 240–243, Washington, DC, USA, 2003. IEEE Computer Society.
[29] Wayne Goddard, Stephen T. Hedetniemi, David P. Jacobs, and Pradip K. Srimani.
Self-stabilizing global optimization algorithms for large network graphs. International Journal of Distributed Sensor Networks, 1(3–4):329–344, 2010.
73
[30] Wayne Goddard, Stephen T. Hedetniemi, David P. Jacobs, Pradip K. Srimani, and Zhenyu Xu. Self-stabilizing graph protocols. Parallel Processing Letters, 18(1):189–
199, 2008.
[31] Wayne Goddard, Stephen T. Hedetniemi, David P. Jacobs, and Vilmar Trevisan.
Distance-k knowledge in self-stabilizing algorithms. Theoretical Computer Science, 399(1–2):118–127, 2008.
[32] Wayne Goddard, Stephen T. Hedetniemi David P. Jacobs, and Pradip K Srimani.
Self-stabilizing protocols for maximal matching and maximal independent sets for ad hoc networks. In Proceedings of the 5th IPDPS Workshop on Advances in Parallel and Distributed Computational Models, WAPDCM’03, pages 162–167. IEEE Computer Society, 2003.
[33] Wayne Goddard and Pradip K. Srimani. Anonymous self-stabilizing distributed algo-rithms for connected dominating set in a network graph. In Proceedings of the 1st In-ternational Multi-Conference on Complexity, Informatics, and Cybernetics, IMCIC’10, 2010.
[34] Nabil Guellati and Hamamache Kheddouci. A survey on self-stabilizing algorithms for independent, domination, coloring, and matching in graphs. Journal of Parallel and Distributed Computing, 70(4):406–415, 2010.
[35] Teresa W. Haynes, Stephen T. Hedetniemi, and Peter J. Slater. Fundamentals of Domi-nation in Graphs. Monographs and Textbooks in Pure and Applied Mathematics. CRC Press, 1998.
[36] Sandra M. Hedetniemi, Stephen T. Hedetniemi, David P. Jacobs, and Pradip K. Sri-mani. Self-stabilizing algorithms for minimal dominating sets and maximal independent sets. Computer Mathematics and Applications, 46(5–6):805–811, 2003.
[37] Stephen T. Hedetniemi, David P. Jacobs, and Pradip K. Srimani. Maximal matching stabilizes in time O(m). Information Processing Letters, 80(5):221–223, December 2001.
[38] Stephen T. Hedetniemi, David P. Jacobs, and Pradip K. Srimani. Linear time self-stabilizing colorings. Information Processing Letters, 87(5):251–255, September 2003.
[39] Lisa Higham and Zhiying Liang. Self-stabilizing minimum spanning tree construction on message-passing networks. In Distributed Computing, volume 2180 of Lecture Notes in Computer Science. Springer-Verlag, October 2001.
[40] Su-Chu Hsu and Shing-Tsaan Huang. A self-stabilizing algorithm for maximal match-ing. Information Processing Letters, 43(2):77–81, August 1992.
[41] Shing-Tsaan Huang, Su-Shen Hung, and Chi-Hung Tzeng. Self-stabilizing coloration in anonymous planar networks. Information Processing Letters, 95(1):307–312, July 2005.
[42] Shing-Tsaan Huang and Chi-Hung Tzeng. Distributed edge coloration for bipartite networks. In Proceedings of the 8th International Conference on Stabilization, Safety, and Security of Distributed Systems, SSS’06, pages 363–377, Berlin, Heidelberg, 2006.
Springer-Verlag.
[43] Tetz C. Huang, Chih-Yuan Chen, and Cheng-Pin Wang. A linear-time self-stabilizing algorithm for the minimal 2-dominating set problem in general networks. Journal of Information Science and Engineering, 24(1):175–187, 2008.
[44] Tetz C. Huang, Ji-Cherng Lin, Chih-Yuan Chen, and Cheng-Pin Wang. A self-stabilizing algorithm for finding a minimal 2-dominating set assuming the distributed demon model. Computers and Mathematics with Applications, 54(3):350–356, August 2007.
75
[45] Michiyo Ikeda, Sayaka Kamei, and Hirotsugu Kakugawa. A space-optimal self-stabilizing algorithm for the maximal independent set problem. In Proceedings of the 3rd International Conference on Parallel and Distributed Computing, Applications and Technologies, PDCAT’02, pages 70–74, 2002.
[46] Alon Itai and Michael Rodeh. Symmetry breaking in distributed networks. Information and Computation, 88:150–158, 1981.
[47] S. Jahanbekam. A comment to: Two classes of edge domination in graphs. Discrete Applied Mathematics, 157(2):400–401, 2009.
[48] Ankur Jain and Arobinda Gupta. A distributed self-stabilizing algorithm for finding a connected dominating set in a graph. In Proceedings of the 6th International Confer-ence on Parallel and Distributed Computing Applications and Technologies, PDCAT’05, pages 615–619, Washington, DC, USA, 2005. IEEE Computer Society.
[49] Hirotsugu Kakugawa and Toshimitsu Masuzawa. A self-stabilizing minimal dominating set algorithm with safe convergence. In Proceedings of the 20th International Conference on Parallel and Distributed Processing, IPDPS’06, pages 263–263, Washington, DC, USA, 2006. IEEE Computer Society.
[50] Sayaka Kamei and Hirotsugu Kakugawa. A self-stabilizing algorithm for the distributed minimal k-redundant dominating set problem in tree networks. In Proceedings of the 4th International Conference on Parallel and Distributed Computing, Applications and Technologies, PDCAT’03, pages 720–724, Washington, DC, USA, 2003. IEEE.
[51] Sayaka Kamei and Hirotsugu Kakugawa. A self-stabilizing approximation algorithm for the distributed minimum k-domination. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E88-A(5):1109–1116, May 2005.
[52] Mehmet Hakan Karaata. Self-stabilizing strong fairness under weak fairness. IEEE Transactions on Parallel and Distributed Systems, 12(4):337–345, April 2001.
[53] Adrian Kosowski and Lukasz Kuszner. Self-stabilizing algorithms for graph coloring with improved performance guarantees. In Proceedings of the 8th International Con-ference on Artificial Intelligence and Soft Computing, ICAISC’06, pages 1150–1159, Berlin, Heidelberg, 2006. Springer-Verlag.
[54] Fabian Kuhn, Thomas Moscibroda, and Rogert Wattenhofer. What cannot be com-puted locally! In Proceedings of the 23rd Annual ACM Symposium on Principles of Distributed Computing, PODC’04, pages 300–309, New York, NY, USA, 2004. ACM.
[55] Ji-Cherng Lin and Tetz C. Huang. An efficient fault-containing self-stabilizing algorithm for finding a maximal independent set. IEEE Transactions on Parallel and Distributed Systems, 14(8):742–754, August 2003.
[56] Nathan Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1):193–201, February 1992.
[57] Fredrik Manne and Morten Mjelde. A self-stabilizing weighted matching algorithm. In Proceedings of the 9th International Conference on Stabilization, Safety, and Security of Distributed Systems, SSS’07, pages 383–393, Berlin, Heidelberg, 2007. Springer-Verlag.
[58] Fredrik Manne, Morten Mjelde, Laurence Pilard, and S´ebastien Tixeuil. A new self-stabilizing maximal matching algorithm. In Proceedings of the 14th International Conference on Structural Information and Communication Complexity, SIROCCO’07, pages 96–108, Berlin, Heidelberg, 2007. Springer-Verlag.
[59] Toshimitsu Masuzawa and S´ebastien Tixeuil. A self-stabilizing link-coloring protocol resilient to unbounded Byzantine faults in arbitrary networks. In JamesH. Anderson,
77
Giuseppe Prencipe, and Roger Wattenhofer, editors, Principles of Distributed Systems, volume 3974 of Lecture Notes in Computer Science, pages 118–129. Springer Berlin Heidelberg, 2006.
[60] Brendan D. McKay and Cheryl E. Praeger. Vertex-transitive graphs which are not cayley graphs, I. Journal of the Australian Mathematical Society (Series A), 56:53–63, February 1994.
[61] A. Meir and J. W. Moon. Relations between packing and covering numbers of a tree.
Pacific Journal of Mathematics, 61(1):225–233, 1975.
[62] Mikhail Nesterenko and Anish Arora. Stabilization-preserving atomicity refinement.
Journal of Parallel and Distributed Computing, 62(5):766–791, May 2002.
[63] Reuay-Ching Pan, Jone-Zen Wang, and Louis R. Chow. A self-stabilizing distributed spanning tree construction algorithm with a distributed demon. Tamsui Oxford Journal of Mathematical Sciences, 15:23–32, May 1999.
[64] R. Saei and S.M. Sheikholeslami. Signed star k-subdomination numbers in graphs.
Discrete Applied Mathematics, 156(15):3066–3070, 2008.
[65] Marco Schneider. Self-stabilization. ACM Computing Surveys, 25(1):45–67, 1993.
[66] Zhengnan Shi, Wayne Goddard, and Stephen T. Hedetniemi. An anonymous self-stabilizing algorithm for 1-maximal independent set in trees. Information Processing Letters, 91(2):77–83, 2004.
[67] Sandeep K. Shukla, Daniel J. Rosenkrantz, and S. S. Ravi. Observations on self-stabilizing graph algorithms for anonymous networks. In Proceedings of the 2nd Work-shop on Self-Stabilizing Systems, WSS’95, pages 7.1–7.15, 1995.
[68] Sandeep K. Shukla, Daniel J. Rosenkrantz, and S. S. Ravi. Simulation and validation tool for self-stabilizing protocols. In Proceedings of 2nd SPIN Workshop, Volume 32 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 1997.
[69] Pradip K. Srimani and Zhenyu Xu. Self-stabilizing algorithms of constructing spanning tree and weakly connected minimal dominating set. In Proceedings of the 27th Interna-tional Conference on Distributed Computing Systems Workshops, ICDCSW’07, pages 3–11, Washington, DC, USA, 2007. IEEE Computer Society.
[70] Huang Sun, Brice Effantin, and Hamamache Kheddouci. A self-stabilizing algorithm for the minimum color sum of a graph. In Proceedings of the 9th International Con-ference on Distributed Computing and Networking, ICDCN’08, pages 209–214, Berlin, Heidelberg, 2008. Springer-Verlag.
[71] Sumit Sur and Pradip K Srimani. A self-stabilizing distributed algorithm to construct bfs spanning trees of a symmetric graph. Computers and Mathematics with Applications, 30, 1992.
[72] Sumit Sur and Pradip K. Srimani. A self-stabilizing algorithm for coloring bipartite graphs. Information Sciences, 69(3):219–227, April 1993.
[73] Ming-Shin Tsai and Shing-Tsaan Huang. A self-stabilizing algorithm for the shortest paths problem with a fully distributed demon. Parallel Processing Letters, 4(1–2):65–72, June 1994.
[74] Shihyu Tsai. An efficient self-stabilizing algorithm for the minimal dominating set problem under a distributed scheduler. Master’s thesis, Department of Applied Math-ematics, National Chiao Tung University, Hsinchu, Taiwan, 2011.
79
[75] Volker Turau. Linear self-stabilizing algorithms for the independent and dominating set problems using an unfair distributed scheduler. Information Processing Letters, 103(3):88–93, 2007.
[76] Volker Turau. Efficient transformation of distance-2 self-stabilizing algorithms. Journal of Parallel Distribted Computing, 72(4):603–612, 2012.
[77] Chi-Hung Tzeng, Jehn-Ruey Jiang, and Shing-Tsaan Huang. A self-stabilizing (δ + 4)-edge-coloring algorithm for planar graphs in anonymous uniform systems. Information Processing Letters, 101(4):168–173, February 2007.
[78] Changping Wang. The signed star domination numbers of the cartesian product graphs.
Discrete Applied Mathematics, 155(11):1497–1505, 2007.
[79] Douglas B. West. Introduction to Graph Theory. Prentice-Hall, Inc., 2nd edition, 2000.
[80] Baogen Xu. On signed edge domination numbers of graphs. Discrete Mathematics, 239(1–3):179–189, 2001.
[81] Baogen Xu. On edge domination numbers of graphs. Discrete Mathematics, 294(3):311–
316, 2005.
[82] Baogen Xu. Two classes of edge domination in graphs. Discrete Applied Mathematics, 154(10):1541–1546, 2006.
[83] Zhenyu Xu, Stephen T. Hedetniemi, Wayne Goddard, and Pradip K. Srimani. A syn-chronous self-stabilizing minimal domination protocol in an arbitrary network graph.
In Proceedings of the 5th International Workshop on Distributed Computing, Springer LNCS 2918, IWDC’03, pages 26–32, 2003.