Search

Abstract: Let denote a graph formed from unlabelled graphs and a labelled graph replacing every vertex of by the graph and joining the vertices of with all the vertices of those of whenever . For unlabelled graphs , let stand for the class of all graphs taken over all possible orderings of . A prime inductive class of graphs, , is said to be a set of all graphs, which can be produced by recursive applying of where is a graph from a fixed set of prime graphs and are either graphs from the set of prime graphs or graphs obtained in the previous steps. Similar inductive definitions for cographs, -trees, series–parallel graphs, Halin graphs, bipartite cubic graphs or forbidden structures of some graph classes were considered in the literature (Batagelj (1994) Drgas-Burchardt et al. (2010) and Hajós (1961) ). This paper initiates a study of prime inductive classes of graphs giving a result, which characterizes, in their language, the substitution closed induced hereditary graph classes. Moreover, for an arbitrary induced hereditary graph class it presents a method for the construction of maximal induced hereditary graph classes contained in and substitution closed. The main contribution of this paper is to give a minimal forbidden graph characterization of induced hereditary prime inductive classes of graphs. As a consequence, the minimal forbidden graph characterization for some special induced hereditary prime inductive graph classes is given There is also offered an algebraic view on the class of all prime inductive classes of graphs of the type . [Copyright &y& Elsevier]

Abstract: This work initiates a systematic investigation into the matrix forms of the Pascal triangle as mathematical objects in their own right. The present paper is especially devoted to the so-called G-matrices, i.e. the set of the twelve triangular matrix forms that can be derived from the Pascal triangle expanded to the level . For , the G-matrix set reduces to a set of four distinct matrices. The twelve G-matrices are defined and the classic Pascal recursion is reformulated for each of the twelve G-matrices. Three sets of matrix transformations are then introduced to highlight different relations between the twelve G-matrices and for generating them from appropriately chosen subsets. [Copyright &y& Elsevier]