Graphs with few P4's under the convexity of paths of order three

A graph is ( q , q - 4 ) if every subset of at most q vertices induces at most q - 4 P 4 's. It therefore generalizes some different classes, as cographs and P 4 -sparse graphs. In this work, we propose algorithms for determining various NP-Hard graph convexity parameters within the convexity of paths of order three, for ( q , q - 4 ) graphs. All algorithms have linear-time complexity, for fixed q , and then are fixed parameter tractable. Moreover, we prove that the Caratheodory number is at most three for every cograph, P 4 -sparse graph and every connected ( q , q - 4 ) -graph with at least q vertices.

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