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Juan Alberto Rodríguez-Velázquez, Erick David Rodríguez-Bazan and Alejandro Estrada-Moreno

.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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Michelle Edwards, Gary MacGillivray and Shahla Nasserasr

, London Mathematical Society Lecture Notes Series 409 (Cambridge University Press, 2013) 127–160. [8] T. Ito, E.D. Demaine, N.J.A. Harvey, C.H. Papadimitriou, M. Sideri, R. Uehara and Y. Uno, On the complexity of reconfiguration problems , Theoret. Comput. Sci. 412 (2011) 1054–1065. doi:10.1016/j.tcs.2010.12.005 [9] S.A. Lakshmanan and A. Vijayakumar, The gamma graph of a graph , AKCE Int. J. Graphs Comb. 7 (2010) 53–59. [10] A. Meir and J.W. Moon, Relations between packing and covering numbers of a tree , Pacific J. Math. 61 (1975) 225

Open access

Camino Balbuena, Cristina Dalfó and Berenice Martínez-Barona

References [1] G. Araujo-Pardo, C. Balbuena, L. Montejano and J.C. Valenzuela, Partial linear spaces and identifying codes , European J. Combin. 32 (2011) 344–351. doi:10.1016/j.ejc.2010.10.014 [2] C. Balbuena, C. Dalfó and B. Martínez-Barona, Characterizing identifying codes from the spectrum of a graph or digraph , Linear Algebra Appl. 570 (2019) 138–147. doi:10.1016/j.laa.2019.02.010 [3] C. Balbuena, F. Foucaud and A. Hansberg, Locating-dominating sets and identifying codes in graphs of girth at least 5, Electron. J. Combin. 22

Open access

Jochen Harant and Samuel Mohr

Abstract

For a graph G with vertex set V (G) and independence number α(G), Selkow [A Probabilistic lower bound on the independence number of graphs, Discrete Math. 132 (1994) 363–365] established the famous lower bound vV(G)1d(v)+1(1+max{d(v)d(v)+1-uN(v)1d(u)+1,0}) on α (G), where N(v) and d(v) = |N(v)| denote the neighborhood and the degree of a vertex vV (G), respectively. However, Selkow’s original proof of this result is incorrect. We give a new probabilistic proof of Selkow’s bound here.

Open access

Anna Bień

Abstract

A set SV is a dominating set of a graph G = (V, E) if every vertex υV which does not belong to S has a neighbour in S. The domination number γ(G) of the graph G is the minimum cardinality of a dominating set in G. A dominating set S is a γ-set in G if |S| = γ(G).

Some graphs have exponentially many γ-sets, hence it is worth to ask a question if a γ-set can be obtained by some transformations from another γ-set. The study of gamma graphs is an answer to this reconfiguration problem. We give a partial answer to the question which graphs are gamma graphs of trees. In the second section gamma graphs γ.T of trees with diameter not greater than five will be presented. It will be shown that hypercubes Qk are among γ.T graphs. In the third section γ.T graphs of certain trees with three pendant vertices will be analysed. Additionally, some observations on the diameter of gamma graphs will be presented, in response to an open question, published by Fricke et al., if diam(T (γ)) = O(n)?

Open access

Michael A. Henning and Alister J. Marcon

References [1] M. Blidia, M. Chellali and S. Khelifi, Vertices belonging to all or no minimum double dominating sets in trees , AKCE Int. J. Graphs. Comb. 2 (2005) 1–9. [2] E.J. Cockayne, M.A. Henning and C.M. Mynhardt, Vertices contained in all or in no minimum total dominating set of a tree , Discrete Math. 260 (2003) 37–44. doi:10.1016/S0012-365X(02)00447-8 [3] W. Goddard, M.A. Henning and C.A. McPillan, Semitotal domination in graphs , Util. Math. 94 (2014) 67–81. [4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals

Open access

Monika Rosicka

R eferences [1] A.P. Burger, C.M. Mynhardt and W.D. Weakley, On the domination number of prisms of graphs , Discuss. Math. Graph Theory 2 (2004) 303–318. doi:10.7151/dmgt.1233 [2] G. Chartrand and F. Harary, Planar permutation graphs , Ann. Inst. Henri Poincare 3 (1967) 433–438. [3] M. Lemańska and R. Zuazua, Convex universal fixers , Discuss. Math. Graph Theory 32 (2012) 807–812. doi:10.7151/dmgt.1631 [4] C.M. Mynhardt and Z. Xu, Domination in prisms of graphs: universal fixers , Util. Math. 78 (2009) 185–201. [5] M

Open access

Teresa W. Haynes and Michael A. Henning

R eferences [1] W.J. Desormeaux and M.A. Henning, Paired domination in graphs: A survey and recent results , Util. Math. 94 (2014) 101–166. [2] W. Goddard, M.A. Henning and C.A. McPillan, Semitotal domination in graphs , Util. Math. 94 (2014) 67–81. [3] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). [4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, Inc. New York, 1998). [5] T.W. Haynes and M

Open access

Joanna Cyman, Michael A. Henning and Jerzy Topp

.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

Open access

Terry A. McKee

)90510-Z [5] R.C.S. Machado, C.M.H. de Figueiredo and N. Trotignon, Complexity of colouring problems restricted to unichord-free and { square,unichord } -free graphs , Discrete Appl. Math. 164 (2014) 191–199. doi:10.1016/j.dam.2012.02.016 [6] T.A. McKee, Independent separator graphs , Util. Math. 73 (2007) 217–224. [7] T.A. McKee, A new characterization of unichord-free graphs , Discuss. Math. Graph Theory 35 (2015) 765–771. doi:10.7151/dmgt.1831 [8] T.A. McKee and F.R. McMorris, Topics in Intersection Graph Theory (Society for