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

TL;DR

This paper presents fixed-parameter tractable algorithms with linear time complexity for finding dominating sets of fixed size in degenerated graphs and related graph classes, improving efficiency over previous methods.

Contribution

The paper introduces new linear-time fixed-parameter algorithms for dominating set problems in degenerated, $K_h$-minor-free, and topological minor-free graphs, extending known results beyond planar graphs.

Findings

Algorithms run in $k^{O(dk)} n$ time for $d$-degenerated graphs.

Improved algorithms with $(O(h))^{hk} n$ and $(O( ext{log } h))^{hk/2} n$ time for $K_h$-topological minor-free and minor-free graphs.

Induced cycle detection in $H$-minor-free graphs can be done in linear or near-linear time.

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.