# Vertex partitioning problems on partial *k*-trees

## Abstract

We describe a general approach to obtain polynomial-time algorithms over partial *k*-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 *k*-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 *k*-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 *k*-tree can be partitioned. (A co-dominating set of a graph is a dominating set of its complement graph).

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## References

- [ALS91]S. Arnborg, J. Lagergren, and D. Seese. Easy problems for tree decomposable graphs.
*J. Algorithms*, 12:308–340, 1991.CrossRefGoogle Scholar - [Arn85]S. Arnborg. Efficient algorithms for combinatorial problems on graphs with bounded decomposability.
*BIT*, 25:2–33, 1985.Google Scholar - [BJ93]H.L. Bodlaender and K. Jansen. On the complexity of scheduling incompatible jobs with unit-times. In
*Lecture Notes in Computer Science (Proc.*18th*MFCS)*, volume 711, pages 291–300. Springer-Verlag, 1993.Google Scholar - [BLW87]M.W. Bern, E.L. Lawler, and A.L. Wong. Linear-time computation of optimal subgraphs of decomposable graphs.
*J. Algorithms*, 8:216–235, 1987.CrossRefGoogle Scholar - [Bod88]H.L. Bodlaender. Dynamic programming on graphs with bounded treewidth. In
*Lecture Notes in Computer Science (Proc. 15th ICALP)*, volume 317, pages 105–119. Springer-Verlag, 1988.Google Scholar - [Bod93]H.L. Bodlaender. A linear time algorithm for finding tree-decompositions of small treewidth. In
*Proc.*25th*STOC*, pages 226–234, 1993.Google Scholar - [BPT92]R.B. Borie, R.G. Parker, and C.A. Tovey. Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families.
*Algorithmica*, 7:555–581, 1992.CrossRefGoogle Scholar - [Cou90a]B. Courcelle. Graph rewriting: an algebraic and logic approach. In J. van Leeuwen, editor,
*Handbook of Theoretical Computer Science*, volume B, pages 193–242. Elsevier, Amsterdam, 1990.Google Scholar - [Cou90b]B. Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs.
*Information and Computation*, 85:12–75, 1990.CrossRefGoogle Scholar - [Cou91]B. Courcelle. The monadic second-order logic of graphs. V. On closing the gap between definability and recognizability.
*Theoret. Comput. Sci.*, 80:153–202, 1991.CrossRefGoogle Scholar - [GJ79]M.R. Garey and D.S. Johnson.
*Computers and Intractability: A Guide to the Theory of NP-Completeness*. W.H. Freeman and Company, New York, 1979.Google Scholar - [GS84]F. Gécseg and M. Steinby.
*Tree Automata*. Akadémiai Kiadó, Budapest, 1984.Google Scholar - [HU79]J.E. Hopcroft and J.D. Ullman.
*Introduction to Automata Theory, Languages, and Computation*. Addison-Wesley, 1979.Google Scholar - [Kal96]D. Kaller. Definability equals recognizability of partial 3-trees, 1996. To appear.Google Scholar
- [KGS95a]D. Kaller, A. Gupta, and T. Shermer. The
*Xt*-coloring problem. In*Lecture Notes in Computer Science (Proc.*12th*STACS)*, volume 900, pages 409–420. Springer-Verlag, 1995.Google Scholar - [KGS95b]D. Kaller, A. Gupta, and T. Shermer. Regular-factors in the complements of partial
*k*-trees. In*Lecture Notes in Computer Science (Proc. 4th WADS)*, volume 955, pages 403–414. Springer-Verlag, 1995.Google Scholar - [KH78]D.G. Kirkpatrick and P. Hell. On the complexity of a generalized matching problem. In
*Proc. 10th STOC*, pages 240–245, 1978.Google Scholar - [Lag94]J. Lagergren, October 1994. Personal communication.Google Scholar
- [MP94]S. Mahajan and J.G. Peters. Regularity and locality in
*k*-terminal graphs.*Disc. Appl. Math.*, 54:229–250, 1994.Google Scholar - [RS86]N. Robertson and P.D. Seymour. Graph minors. II. Algorithmic aspects of tree-width.
*J. Algorithms*, 7:309–322, 1986.CrossRefGoogle Scholar - [Tho90]W. Thomas. Automata on infinite objects. In J. van Leeuwen, editor,
*Handbook of Theoretical Computer Science*, volume B, pages 133–191. Elsevier, Amsterdam, 1990.Google Scholar - [WW89]E. Wanke and M. Wiegers. Undecidability of the bandwidth problem on linear graph languages.
*Inform. Process. Lett.*, 33:193–197, 1989.Google Scholar