Polynomial Time Learnability Of Graph Pattern Languages Defined By Cographs

A cograph (complement reducible graph) is a graph which can be generated by disjoint union and complement operations on graphs, starting with a single vertex graph. Cographs arise in many areas of computer science and are studied extensively. With the goal of developing an effective data mining method for graph structured data, in this paper we introduce a graph pattern expression, called a cograph pattern, which is a special type of cograph having structured variables. Firstly, we show that a problem whether or not a given cograph pattern g matches a given cograph G is NP-complete. From this result, we consider the polynomial time learnability of cograph pattern languages defined by cograph patterns having variables labeled with mutually different labels, called linear cograph patterns. Secondly, we present a polynomial time matching algorithm for linear cograph patterns. Next, we give a polynomial time algorithm for obtaining a minimally generalized linear cograph pattern which explains given positive data. Finally, we show that the class of linear cograph pattern languages is polynomial time inductively inferable from positive data.