For r>=3, a clique-extension of order r+1 is a connected graph that consists of a Kr plus another vertex adjacent to at most r-1 vertices of Kr. In this paper we consider the problem of finding the smallest number t such that any graph G of order n admits a decomposition into edge disjoint copies of a fixed graph H and single edges with at most t elements. Here we solve the case when H is a fixed clique-extension of order r+1, for all r>=3 and will also obtain all extremal graphs. This work extends results proved by Bollob{\'a}s [Math.\ Proc.\ Cambridge Philosophical Soc 79 (1976) 19--24] for cliques.

}, url = {http://www.combinatorialmath.ca/arscombinatoria/vol100.html}, attachments = {https://docentes.fct.unl.pt/sites/default/files/tmjs/files/2011-04-clique_extension.pdf}, author = {Sousa, Teresa} }