Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs

Abstract

There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a k O(dk) n time algorithm for finding a dominating set of size at most k in a d-degenerated graph with n vertices. This proves that the dominating set problem is fixed-parameter tractable for degenerated graphs. For graphs that do not contain K h as a topological minor, we give an improved algorithm for the problem with running time (O(h)) hk n. For graphs which are K h -minor-free, the running time is further reduced to (O(log h)) hk/2 n. Fixed-parameter tractable algorithms that are linear in the number of vertices of the graph were previously known only for planar graphs.For the families of graphs discussed above, the problem of finding an induced cycle of a given length is also addressed. For every fixed H and k, we show that if an H -minor-free graph G with n vertices contains an induced cycle of size k, then such a cycle can be found in O(n) expected time as well as in O(n log n) worst-case time. Some results are stated concerning the (im)possibility of establishing linear time algorithms for the more general family of degenerated graphs.

Parameterized Complexity of Directed Steiner Tree on Sparse Graphs

“…Our algorithm for Dominating Set on d-degenerate graphs improves over the O * (k O(dk) ) time algorithm by Alon and Gutner [2]. In fact, it turns out that our algorithm is essentially optimal -we show that, assuming the ETH, the running time dependence of our algorithm on the degeneracy of the input graph and solution size k cannot be significantly improved.…”

“…It follows from the reduction presented in [34] that DST is W [2]-hard on general digraphs. Hence we do not expect FPT algorithms to exist for these problems, and so we turn our attention to classes of sparse digraphs.…”

“…4. DST is W [2]-hard on 2-degenerate digraphs if the graph induced on terminals is allowed to contain directed cycles. 5.…”

“…We claim that these two sets together cover the elements in U (i,j 1 )(i,j 2 ) . But this is true since F ij 1 uv covers the elements of U (i,j 1 )(i,j 2 ) which do not correspond to id(u) and F ij 2 uv covers the elements of U (i,j 1 )(i,j 2 ) which do correspond to id(u). This completes the proof of the lemma.…”

“…For an example the initial study of parameterized subexponential algorithms for Dominating Set, on planar graphs [1,17] resulted in the development of bidimensionality theory characterizing a broad range of graph problems that admit efficient approximation schemes, subexponential time FPT algorithms and efficient polynomial time pre-processing (called kernelization) on minor closed graph classes [10,11]. Alon and Gutner [2] and Philip, Raman, and Sikdar [37] showed that Dominating Set problem is FPT on graphs of bounded degeneracy and on K i,j -free graphs, respectively.…”