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8 (1956), 399–404. , HOPCROFT, J. , AND WYLLIE, J. The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10 (1980), 111–121. [19] GAREY, M. , AND JOHNSON, D. S. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, NY, 1979. [20] HAAS, R. Service Deployment in Programmable Networks. PhD thesis, ETH Zurich, Switzerland, 2003. [21] KARP, R. M. On the complexity of combinatorial problems. Networks 5 (1975), 45–68. [22] LUEKER, G. , ROSE, D. , AND TARJAN, R.

So far, such a linear time algorithm has been constructed (see [3,4]) only for the parameters of treewidth and pathwidth (actually, the result in [3,4] can be directly transfered to the node search number which is known to be equal to the pathwidth plus one – see [12,13,17]). To be precise, [3,4] state that for fixed one can determine in linear time whether a given graph has pathwidth at most and if so, find a path decomposition of minimum width. This algorithm first finds a tree decomposition of width at most if it exists (if not, then the pathwidth is also larger than and then uses this tree decomposition to solve the problem, using the result in [4] that states that for fixed and one can test whether the pathwidth of a graph G is at most and if so, find a minimum width path Computing Small Search Numbers in Linear Time 39 decomposition, assuming that G is given together with a tree decomposition of width at most This paper uses a similar idea: we first determine a path decomposition of bounded width (if such a path decomposition does not exist, we know that the linear-width is also not bounded), and then apply the main algorithm, presented in this paper, that, given such a path decomposition, solves our problem.

11. H. A. Kierstead, W. A. Trotter: An Extremal Problem in Recursive Combinatorics. Congressus Numeratium 33, pp 143-153, 1981. 12. N. G. Kinnersley: The Vertex Separation Number of a Graph equals its Path-Width. Information processing Letters 42(6), pp. 345-350, 1992. 13. L. M. Kirousis, D. M. Thilikos: The Linkage of a Graph. SIAM Journal on Computing 25(3), pp. 626-647, 1996. 14. L. Lovasz, M. E. Saks, W. A. Trotter: An On-Line Graph Coloring Algorithm with Sublinear Performance Ratio. TM-ARH-013-014.

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