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Let G = (V, E) be a graph of order n = |V| and chromatic number (G) A dominating set D of G is called a dominating chromatic partition-cover or dcc-set, if it intersects every color class of every X-coloring of G. The minimum cardinality of a dcc-set is called the dominating chromatic partition-covering number, denoted dcc(G). The dcc-saturation number equals the minimum integer i such that every vertex ν ∈ V is contained in a dcc-set of cardinality k.This number is denoted by dccs(G) In this paper we study a few properties ofthese two invariants dcc(G) and dccs(G).

Let G be a fuzzy graph. Then D Ôèå V is said to be a strong (weak) fuzzy dominating set of G if every vertex v ∈ V -D is strongly (weakly) dominated by some vertex u in D. We denote a strong (weak) fuzzy dominating set by sfd-set (wfd-set). The minimum scalar cardinality of a sfd-set (wfd-set) is called the strong (weak) fuzzy domination number of G and it is denoted by γsf (G)γwf (G). In this paper we introduce the concept of strong (weak) domination in fuzzy graphs and obtain some interesting results for this new parameter in fuzzy graphs., {"references":["E.J. Cockayne and S.T. Hedetniemi,Towards a theory of domination in\ngraphs, Networks,(1977), 247-261.","Domke et. al,On parameters related to strong and weak domination in\ngraphs,Discrete Mathematics 258 (2002), 1-11.","T.W.Haynes et. al, Fundamentals of Domination in graphs, Marcel\nDekker, New York, 1998.","J.N. Mordeson and P.S. Nair, Fuzzy graphs and Fuzzy Hypergraphs,\nPhysica Verlag, Heidelberg, 1998; second edition 2001.","A. Rosenfeld Fuzzy graphs,in; L.A. Zadeh, K.S. Fu, M.Shimura(Eds.),\nFuzzy sets and their Applications to Cognitive and Decision Processes,\nAcademic Press, New York, 1975, 77-95.","E. Sampathkumar, L. PushpalathaStrong, weak domination and domination\nbalance in a graph, Discrete Math. 161(1996), 235-242.","A. Somasundaram and S. Somasundaram,Domination in fuzzy graphs-I,\nPattern recognition Letters 19(1998), 787-791."]}

In this paper, we present a framework to determine Haar solutions of Bratu-type equations that are widely applicable in fuel ignition of the combustion theory and heat transfer. The method is proposed by applying Haar series for the highest derivatives and integrate the series. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. The results show that the proposed way is quite reasonable when compared to exact solution., {"references":["U.M. Ascher, R. Matheij, R.D. Russell, Numerical solution of boundary\nvalue problems for ordinary differential equations, SIAM, Philadelphia,\nPA, 1995.","J.P. Boyd, Chebyshev polynomial expansions for simultaneous approximation\nof two branches of a function with application to the onedimensional\nBratu equation, Applied Mathematics and Computation 142\n(2003) 189-200.","J.P. Boyd, An analytical and numerical study of the two-dimensional\nBratu equation, Journal of Scientific Computing 1 (2) (1986) 183-206.","R. Buckmire, Investigations of nonstandard Mickens-type finitedifference\nschemes for singular boundary value problems in cylindrical\nor spherical coordinates, Numerical Methods for partial Differential\nequations 19 (3) (2003) 380-398.","R. Buckmire, Application of Mickens finite-difference scheme to the\ncylindrical Bratu Gelfand problem, doi:10.1002/num.10093.","C.F.Chen, C.H.Hsiao, Haar wavelet method for solving lumped and\ndistributed- parameter systems, IEE Proc.Pt.D 144(1) (1997) 87-94.","D.A. Frank-Kamenetski, Diffusion and Heat Exchange in Chemical\nKinetics, Princeton University Press, Princeton, NJ, 1955.","I.H.A.H. Hassan, V.S. Erturk, Applying differential transformation\nmethod to the One-dimensional planar Bratu problem, International\nJournal of Contemporary Mathematical Sciences 2 (2007) 1493-1504.","Hikmet Caglar, Nazan Caglar, Mehmet zer, Antonios Valaristo, Amalia\nN. Miliou, Antonios N. Anagnostopoulos, Dynamics of the solution of\nBratu-s Equation, Nonlinear Analysis, (Press).\n[10] C.H.Hsiao, Haar wavelet approach to linear stiff systems, Mathematics\nand Computers in simultion ,Vol 64, 2004, pp.561-567.\n[11] J. Jacobson, K. Schmitt, The Liouville-Bratu-Gelfand problem for radial\noperators, Journal of Differential Equations 184 (2002) 283-298.\n[12] U.Lepik, Numerical solution of evolution equations by the Haar wavelet\nmethod, Applied Mathematics and Computation 185 (2007) 695-704.\n[13] U.Lepik, Numerical solution of differential equations using Haar\nwavelets, Mathematics and Computers in Simulation 68 (2005) 127-143.\n[14] S. Li, S.J. Liao, An analytic approach to solve multiple solutions of a\nstrongly nonlinear problem, Applied Mathematics and Computation 169\n(2005) 854-865.\n[15] A.S. Mounim, B.M. de Dormale, From the fitting techniques to accurate\nschemes for the Liouville Bratu Gelfand problem, Numerical Methods\nfor Partial Differential Equations, doi: 10.1002/num.20116.\n[16] M.I. Syam, A. Hamdan, An efficient method for solving Bratu equations,\nApplied Mathematics and Computation 176 (2006) 704-713.\n[17] A.M. Wazwaz, A new method for solving singular initial value problems\nin the second order differential equations, Applied Mathematics and\nComputation 128 (2002) 47-57.\n[18] A.M. Wazwaz, Adomian decomposition method for a reliable treatment\nof the Bratu-type equations, Applied Mathematics and Computation 166\n(2005) 652-663."]}