REIDENBACH, D. and SCHMID, M.L., 2012. Patterns with bounded treewidth. Language and Automata Theory and Applications; Lecture Notes in Computer Science, 7183, pp. 468 - 479.
We show that any parameter of patterns that is an upper
bound for the treewidth of appropriate encodings of patterns as relational
structures, if restricted to a constant, allows the membership problem
for pattern languages to be solved in polynomial time. Furthermore, we
identify a new such parameter, called the scope coincidence degree.