17.
A graph
G was defined in [16] as
P4-reducible, if no vertex in
G belongs to more than one chordless path on four vertices or
P4. A graph
G is defined in [15] as
P4-sparse if no set of five vertices induces more than one
P4, in
G.
P4-sparse graphs generalize both
P4-reducible and the well known class of
p4-free graphs or
cographs. In an extended abstract in [11] the first author introduced a method using the
modular decomposition tree of a graph as the framework for the resolution of algorithmic problems. This method was applied to the study of
P4-sparse and
extended P4-sparse graphs.
In this paper, we begin by presenting the complete information about the method used in [11]. We propose a unique tree representation of P4-sparse and a unique tree representation of P4-reducible graphs leading to a simple linear recognition algorithm for both classes of graphs. In this way we simplify and unify the solutions for these problems, presented in [16–19]. The tree representation of an n-vertex P4-sparse or a P4-reducible graph is the key for obtaining O(n) time algorithms for the weighted version of classical optimization problems solved in [20]. These problems are NP-complete on general graphs.
Finally, by relaxing the restriction concerning the exclusion of the C5 cycles from P4-sparse and P4-reducible graphs, we introduce the class of the extended P4-sparse and the class of the extendedP4-reducible graphs. We then show that a minimal amount of additional work suffices for extending most of our algorithms to these new classes of graphs. 相似文献