Maximum Induced Forests in Graphs of Bounded Treewidth (original) (raw)

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Keywords: Treewidth, chordal graph, induced forest

Abstract

Given a nonnegative integer ddd and a graph GGG, let fd(G)f_d(G)fd(G) be the maximum order of an induced forest in GGG having maximum degree at most ddd. We seek lower bounds for fd(G)f_d(G)fd(G) based on the order and treewidth of GGG.

We show that, for all k,dge2k,d\ge 2k,dge2 and nge1n\ge 1nge1, if GGG is a graph with order nnn and treewidth at most kkk, then fd(G)gelceil(2dn+2)/(kd+d+1)rceilf_d(G)\ge\lceil{(2dn+2)/(kd+d+1)}\rceilfd(G)gelceil(2dn+2)/(kd+d+1)rceil, unless GinK1,1,3,K2,3G\in\{K_{1,1,3},K_{2,3}\}GinK1,1,3,K2,3 and k=d=2k=d=2k=d=2. We give examples that show that this bound is tight to within 111.

We conjecture a bound for d=1d=1d=1: f_1(G)gelceil2n/(k+2)rceilf_1(G) \ge\lceil{2n/(k+2)}\rceilf1(G)gelceil2n/(k+2)rceil, which would also be tight to within 111, and we prove it for kle3k\le 3kle3. For kge4k\ge 4kge4 the conjecture remains open, and we prove a weaker bound: f1(G)ge(2n+2)/(2k+3)f_1(G)\ge (2n+2)/(2k+3)f_1(G)ge(2n+2)/(2k+3). We also examine the cases d=0d=0d=0 and k=0,1k=0,1k=0,1.

Lastly, we consider open problems relating to fdf_dfd for graphs on a given surface, rather than graphs of bounded treewidth.

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How to Cite

Chappell, G. G., & Pelsmajer, M. J. (2013). Maximum Induced Forests in Graphs of Bounded Treewidth. The Electronic Journal of Combinatorics, 20(4), #P8. https://doi.org/10.37236/3826