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On Accurate Domination in Graphs

.B. Kattimani, Accurate domination in graphs , in: Advances in Domination Theory I, V.R. Kulli (Ed.), (Vishwa International Publications, 2012) 1–8. [13] V.R. Kulli and M.B. Kattimani, Global accurate domination in graphs , Int. J. Sci. Res. Pub. 3 (2013) 1–3. [14] V.R. Kulli and M.B. Kattimani, Connected accurate domination in graphs , J. Comput. Math. Sci. 6 (2015) 682–687. [15] C.M. Mynhardt, Vertices contained in every minimum dominating set of a tree , J. Graph Theory 31 (1999) 163–177. doi:10.1002/(SICI)1097-0118(199907)31:3h163::AID-JGT2i3

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About (k, l)-Kernels, Semikernels and Grundy Functions in Partial Line Digraphs

R eferences [1] M. Aigner, On the linegraph of a directed graph , Math. Z. 102 (1967) 56–61. doi:10.1007/BF01110285 [2] C. Balbuena and M. Guevara, Kernels and partial line digraphs , Appl. Math. Lett. 23 (2010) 1218–1220. doi:10.1016/j.aml.2010.06.001 [3] C. Berge, Graphs, (North Holland, 1985) Chapter 14. [4] E. Boros and V. Gurvich, Perfect Graphs, Kernels and Cores of Cooperative Games (RUTCOR Research Report 12, Rutgers University, 2003). [5] M.A. Fiol, J.L.A. Yebra and I. Alegre, Line digraph iterations and the ( d, k

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On Two Generalized Connectivities of Graphs

R eferences [1] S.B. Akers and B. Krishnamurthy, A group-theoretic model for symmetric interconnection networks , IEEE Trans. Comput. 38 (1989) 555–566. doi:10.1109/12.21148 [2] L. Babai, Automorphism groups, isomorphism, reconstruction , in: Handbook of Combinatorics, R.L. Graham et al . (Ed(s)), (Elsevier, Amsterdam, 1995) 1447–1540. [3] J.-C. Bermond, O. Favaron and M. Maheo, Hamiltonian decomposition of Cayley graphs of degree 4, J. Combin. Theory Ser. B 46 (1989) 142–153. doi:10.1016/0095-8956(89)90040-3 [4] N. Biggs

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Products Of Digraphs And Their Competition Graphs

Abstract

If D = (V, A) is a digraph, its competition graph (with loops) CGl(D) has the vertex set V and {u, v} ⊆ V is an edge of CGl(D) if and only if there is a vertex wV such that (u, w), (v, w) ∈ A. In CGl(D), loops {v} are allowed only if v is the only predecessor of a certain vertex wV. For several products D 1D 2 of digraphs D 1 and D 2, we investigate the relations between the competition graphs of the factors D 1, D 2 and the competition graph of their product D 1D 2.

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A Note on the Thue Chromatic Number of Lexicographic Products of Graphs

Abstract

A sequence is called non-repetitive if none of its subsequences forms a repetition (a sequence r 1 r 2r 2 n such that ri = rn + i for all 1 ≤ in). Let G be a graph whose vertices are coloured. A colouring ϕ of the graph G is non-repetitive if the sequence of colours on every path in G is non-repetitive. The Thue chromatic number, denoted by π(G), is the minimum number of colours of a non-repetitive colouring of G.

In this short note we present two general upper bounds for the Thue chromatic number for the lexicographic product GH of graphs G and H with respect to some properties of the factors. One upper bound is then used to derive the exact values for π(GH) when G is a complete multipartite graph and H an arbitrary graph.

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Bounding the Open k-Monopoly Number of Strong Product Graphs

R eferences [1] D.W. Bange, A.E. Barkauskas and P.J. Slater, Efficient dominating sets in graphs , in: Applications of Discrete Math., R.D. Ringeisen and F.S. Roberts (Ed(s)), (SIAM, Philadelphia, 1988) 189–199. [2] N. Biggs, Perfect codes in graphs , J. Combin. Theory Ser. B 15 (1973) 289–296. doi:10.1016/0095-8956(73)90042-7 [3] C. Dwork, D. Peleg, N. Pippenger and E. Upfal, Fault tolerance in networks of bounded degree , SIAM J. Comput. 17 (1988) 975–988. doi:10.1137/0217061 [4] H. Fernau, J.A. Rodríguez-Velázquez and J

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On Generalized Sierpiński Graphs

.1007/s10623-012-9642-1 [8] S. Gravier, M. Kovše and A. Parreau, Generalized Sierpiński graphs , in: Posters at EuroComb’11, Rényi Institute, Budapest, 2011. http://www.renyi.hu/conferences/ec11/posters/parreau.pdf [9] A.M. Hinz and C.H. auf der Heide, An efficient algorithm to determine all shortest paths in Sierpiński graphs , Discrete Appl. Math. 177 (2014) 111–120. doi:10.1016/j.dam.2014.05.049 [10] A.M. Hinz, S. Klavžar, U. Milutinović and C. Petr, The Tower of Hanoi—Myths and Maths (Birkhäuser/Springer Basel, 2013). [11] A.M. Hinz, S

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A study on the pendant number of graph products

References [1] D. B. West, Introduction to Graph Theory , Prentice Hall of India, New Delhi, 2005. ⇒24 [2] F. Harary, Graph Theory , Narosa Publishing House, New Delhi, 2001. ⇒24 [3] J. K. Sebastian, J. V. Kureethara, Pendant number of graphs, Int. J. Appl. Math. , IJAM 31 , 5 (2018) 679–689. ⇒25, 34, 35 [4] J. K. Sebastian, J. V. Kureethara, N. K. Sudev, C. Dominic, On star decomposition and star number of some graph classes, Int. J. Scien. Res. Mathe. Stati. Sci. , IJSRMSS , 5 , 6 (2018) 81–85. ⇒25 [5] J. K. Sebastian, J

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The Planar Index and Outerplanar Index of Some Graphs Associated to Commutative Rings

(2002) 73–78. [19] A.V. Kelarev, Ring Constructions and Applications (World Scientific, River Edge, NJ, 2002). [20] A.V. Kelarev, Labelled Cayley graphs and minimal automata , Australasian J. Combinatorics 30 (2004) 95–101. [21] A.V. Kelarev, On Cayley graphs of inverse semigroups , Semigroup Forum 72 (2006) 411–418. doi:10.1007/s00233-005-0526-9 [22] A.V. Kelarev and C.E. Praeger, On transitive Cayley graphs of groups and semigroups , European J. Combinatorics 24 (2003) 59–72. doi:10.1016/S0195-6698(02)00120-8 [23] A.V. Kelarev

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Orientable ℤ N -Distance Magic Graphs

R eferences [1] S. Arumugam, D. Froncek and N. Kamatchi, Distance Magic Graphs—A Survey , J. Indones. Math. Soc., Special Edition (2011) 11–26. doi:10.22342/jims.0.0.15.11-26 [2] G.S. Bloom and D.F. Hsu, On graceful digraphs and a problem in network addressing , Congr. Numer. 35 (1982) 91–103. [3] G.S. Bloom, A. Marr and W.D. Wallis, Magic digraphs , J. Combin. Math. Combin. Comput. 65 (2008) 205–212. [4] S. Cichacz, Note on group distance magic graphs G [ C 4 ], Graphs Combin. 30 (2014) 565–571. doi:10.1007/s00373-013-1294-z

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