We describe a general approach to obtain polynomial-time algorithms over partial fc-trees for graph problems in which the vertex set is to be partitioned in some way. We encode these problems with formulae of the Extended Monadic Second-order (or EMS) logic. Such a formula can be translated into a polynomial-time algorithm automatically. We focus on the problem of partitioning a partial fc-tree into induced subgraphs isomorphic to a fixed pattern graph; a distinct algorithm is derived for each pattern graph and each value of k. We use a “pumping lemma” to show that (for some pattern graphs) this problem cannot be encoded in the “ordinary” Monadic Second-order logic—from which a linear-time algorithm over partial fc-trees would be obtained. Hence, an EMS formula is in some sense the strongest possible. As a further application of our general approach, we derive a polynomial-time algorithm to determine the maximum number of co-dominating sets into which the vertices of a partial fc-tree can be partitioned. (A co-dominating set of a graph is a dominating set of its complement graph).
CITATION STYLE
Gupta, A., Kaller, D., Mahajan, S., & Shermer, T. (1996). Vertex partitioning problems on partial k-trees. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1097, pp. 161–172). Springer Verlag. https://doi.org/10.1007/3-540-61422-2_129
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