Large induced acyclic and outerplanar subgraphs of low-treewidth planar graphs
نویسندگان
چکیده
Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighter results for planar graphs of low treewidth. We show that the Albertson-Berman conjecture holds, and is tight, for planar graphs of treewidth 3 (and, in fact, for any graph of treewidth at most 3). We show that every 2-outerplanar graph has an induced outerplanar graph on at least two-thirds of its vertices. We also show that every 2-outerplanar graph has an induced forest on at least half the vertices by showing that its vertices can be partitioned into two sets, each of which induces a forest.
منابع مشابه
Large Induced Outerplanar and Acyclic Subgraphs of Planar Graphs
Albertson and Berman [1] conjectured that every planar graph has an induced forest on half of its vertices; the current best result, due to Borodin [3], is an induced forest on two fifths of the vertices. We show that the Albertson-Berman conjecture holds, and is tight, for planar graphs of treewidth 3 (and, in fact, for any graph of treewidth at most 3). We also improve on Borodin’s bound for ...
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