On classes of graphs with strongly sublinear separators
نویسنده
چکیده
For real numbers c, ε > 0, let Gc,ε denote the class of graphs G such that each subgraph H of G has a balanced separator of order at most c|V (H)|1−ε. A class G of graphs has strongly sublinear separators if G ⊆ Gc,ε for some c, ε > 0. We investigate properties of such graph classes, leading in particular to an approximate algorithm to determine membership in Gc,ε: there exist c > 0 such that for each input graphG, this algorithm in polynomial time determines either that G ∈ Gc′,ε2/160, or that G 6∈ Gc,ε. A balanced separator in a graph G is a set C ⊆ V (G) such that each component of G−C has at most 2 3 |V (G)| vertices (the constant 2 3 is customary but basically arbitrary, any constant smaller than 1 would give qualitatively the same results). Balanced separators of small order are of obvious importance in the construction of efficient divide-and-conquer style algorithms [13]. In these applications, all subgraphs that appear throughout the recursion are required to have small balanced separators. This motivates us to ask which graphs admit small balanced separators in all their subgraphs. For real numbers c > 0 and 0 < ε ≤ 1, let Gc,ε denote the class of graphs G such that each subgraph H of G has a balanced separator of order at most c|V (H)|1−ε. A class G of graphs has strongly sublinear separators if G ⊆ Gc,ε for some c, ε > 0. Note that if c′ ≤ c and ε′ ≥ ε, then Gc′,ε′ ⊆ Gc,ε. Possibly the best known example of a class with strongly sublinear separators is the class of planar graphs—Lipton and Tarjan [12] proved that all planar graphs belong to G8,1/2 (the constant 1/2 is tight as shown by planar grids, the constant √ 8 has been subsequently improved [3]). More generally, Gilbert et al. [10] proved that for every surface Σ there exists ∗Computer Science Institute, Charles University, Prague, Czech Republic. E-mail: [email protected]. Supported by the Center of Excellence – Inst. for Theor. Comp. Sci., Prague (project P202/12/G061 of Czech Science Foundation).
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ورودعنوان ژورنال:
- CoRR
دوره abs/1710.03117 شماره
صفحات -
تاریخ انتشار 2017