2004/01/01 by Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi +1 · 111 citations
Computer Science · Mathematics · #1-planar graph #Advanced Graph Theory Research #Bounded function #Combinatorics #Complexity and Algorithms in Graphs #Discrete mathematics #Graph #Graph minor #Graph power #Line graph #Mathematics #Optimization and Search Problems #Partial k-tree #Pathwidth #Treewidth
paper · doi:10.1137/s0895480103433410
published in SIAM Journal on Discrete Mathematics 18(3), 501-511 (Society for Industrial and Applied Mathematics)
openalex publication_date 2004/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
For several graph-theoretic parameters such as vertex cover and dominating set, it is known that if their sizes are bounded by k, then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixed-parameter algorithms on minor-closed graph classes such as planar graphs, single-crossing-minor-free graphs, and graphs of bounded genus. In this paper we examine whether similar bounds can be obtained for larger minor-closed graph classes and for general families of graph parameters, including all those for which such behavior has been reported so far. Given a graph parameter P, we say that a graph family F has the parameter-treewidth property for P if there is an increasing function t such that every graph G\inF has treewidth at most t(P(G)). We prove as our main result that, for a large family of graph parameters called contraction-bidimensional, a minor-closed graph family F has the parameter-treewidth property if F has bounded local treewidth. We also show "if and only if" for some graph parameters, and thus, this result is in some sense tight. In addition we show that, for a slightly smaller family of graph parameters called minor-bidimensional, all minor-closed graph families F, excluding some fixed graphs, have the parameter-treewidth property. The contraction-bidimensional parameters include many domination and covering graph parameters such as vertex cover, feedback vertex set, dominating set, edge-dominating set, and q-dominating set (for fixed q). We use our theorems to develop new fixed-parameter algorithms in these contexts.