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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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Characterizing Atoms that Result from Decomposition by Clique Separators

)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

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Removable Edges on a Hamilton Cycle or Outside a Cycle in a 4-Connected Graph

. Theory Ser. B 17 (1974) 281–298. doi:10.1016/0095-8956(74)90034-3 [5] J.C. Wu, H.J. Broersma and H. Kang, Removable edges and chords of longest cycles in 3 -connected graphs , Graphs Combin. 30 (2014) 743–753. doi:10.1007/s00373-013-1296-x [6] J.C. Wu, X.L. Li and L.S. Wang, Removable edges in a cycle of a 4 -connected graph , Discrete Math. 287 (2004) 103–111. doi:10.1016/j.disc.2004.05.015 [7] J.C.Wu, X.L. Li and J.J. Su, The number of removable edges in a 4 -connected graph , J. Combin. Theory Ser. B 92 (2004) 13–40. doi:10.1016/j

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The {−2,−1}-Selfdual and Decomposable Tournaments

R eferences [1] M. Achour, Y. Boudabbous and A. Boussaïri, The {−3} -reconstruction and the {−3} -self duality of tournaments , Ars Combin. 122 (2015) 355–377. [2] M. Basso-Gerbelli and P. Ille, La reconstruction des relations définies par interdits , C. R. Acad. Sci. Paris, Sér. I Math. 316 (1993) 1229–1234. [3] H. Belkhechine, I. Boudabbous and J. Dammak, Morphologie des tournois (−1)- critiques , C. R. Acad. Sci. Paris, Sér. I Math. 345 (2007) 663–666. doi:10.1016/j.crma.2007.11.006 [4] A. Bondy and R.L. Hemminger, Graph

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Strongly Unichord-Free Graphs

R eferences [1] A. Brandstädt, V.B. Le and J.P. Spinrad, Graph Classes: A Survey (Society for Industrial and Applied Mathematics, Philadelphia, 1999). doi:10.1137/1.9780898719796 [2] G.A. Dirac, Minimally 2 -connected graphs , J. Reine Angew. Math. 228 (1967) 204–216. doi:10.1515/crll.1967.228.204 [3] B. Lévêque, F. Maffray and N. Trotignon, On graphs with no induced subdivision of K 4 , J. Combin. Theory Ser. B 102 (2012) 924–947. doi:10.1016/j.jctb.2012.04.005 [4] R.C.S. Machado and C.M.H. de Figueiredo, Total chromatic number of

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A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs

R eferences [1] L.W. Beineke, Derived graphs and digraphs , in: Beitrage zur Graphentheorie, H. Sachs, H.-J. Voss, H.-J. Walter (Ed(s)), (Leipzig, Teubner, 1968) 17–33. [2] J.C. Bermond and J.C. Meyer, Graphs representatif des arêtes d’un multigraphe , J. Math. Pures Appl. 52 (1973) 299–308. [3] V. Chvátal and P.L. Hammer, Set-Packing and Threshold Graphs (Comp. Sci. Dept. Univ. of Waterloo, Ontario, 1973). [4] D.G. Degiorgi and K. Simon, A dynamic algorithm for line graph recognition , Lect. Notes in Comput. Sci. 1017 (1995) 37

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Kaleidoscopic Colorings of Graphs

R eferences [1] M. Aigner, E. Triesch and Zs. Tuza, Irregular assignments and vertex-distinguishing edge-colorings of graphs , Combinatorics’ 90 Elsevier Science Pub., New York (1992) 1–9. [2] A C. Burris and R.H. Schelp, Vertex-distinguishing proper edge colorings , J. Graph Theory 26 (1997) 73–82. doi:10.1002/(SICI)1097-0118(199710)26:2〈73::AID-JGT2〉3.0.CO;2-C [3] J. Černý, M. Horňák and R. Soták, Observability of a graph , Math. Slovaca 46 (1996) 21–31. [4] G. Chartrand. S. English and P. Zhang, Binomial colorings of graphs

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𝒫-Apex Graphs

R eferences [1] J.A. Bondy and U.S.R. Murty, Graph Theory (Springer, 2008). [2] M. Borowiecki and P. Mihók, Hereditary properties of graphs , in: V.R. Kulli, Ed., Advances in Graph Theory (Vishawa International Publication Gulbarga, 1991). [3] S. Cichacz, A. Görlich, M. Zwonek and A. Żak, On ( C n ; k ) stable graphs , Electron J. Combin. 18 (2011) #P205. [4] E. Drgas-Burchardt, Forbidden graphs for classes of split-like graphs , European J. Combin. 39 (2014) 68–79. doi:10.1016/j.ejc.2013.12.004 [5] E. Drgas-Burchardt, On

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The Graphs Whose Permanental Polynomials Are Symmetric

R eferences [1] M. Borowiecki and T. Jóźwiak, Computing the permanental polynomial of a multi-graph , Discuss. Math. 5 (1982) 9–16. [2] M. Borowiecki and T. Jóźwiak, A note on characteristic and permanental polynomials of multigraphs , in: Graph Theory, M. Borowiecki, J.W. Kenendy and M.M. Syslo (Ed(s)), (Springer-Verlag, Berlin, 1983) 75–78. doi:10.1007/bfb0071615 [3] M. Borowiecki, On spectrum and per-spectrum of graphs , Publ. Inst. Math. (Beograd) (N.S.) 38 (1985) 31–33. [4] Q. Chou, H. Liang and F. Bai, Computing the

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Split Euler Tours In 4-Regular Planar Graphs

Abstract

The construction of a homing tour is known to be NP-complete. On the other hand, the Euler formula puts su cient restrictions on plane graphs that one should be able to assert the existence of such tours in some cases; in particular we focus on split Euler tours (SETs) in 3-connected, 4-regular, planar graphs (tfps). An Euler tour S in a graph G is a SET if there is a vertex v (called a half vertex of S) such that the longest portion of the tour between successive visits to v is exactly half the number of edges of G. Among other results, we establish that every tfp G having a SET S in which every vertex of G is a half vertex of S can be transformed to another tfp G′ having a SET S′ in which every vertex of G′ is a half vertex of S′ and G′ has at most one point having a face configuration of a particular class. The various results rely heavily on the structure of such graphs as determined by the Euler formula and on the construction of tfps from the octahedron. We also construct a 2-connected 4-regular planar graph that does not have a SET.

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