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arXiv · 1602.08631

The Second Neighborhood Conjecture for Oriented Graphs Missing Combs

Salman Ghazal
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Abstract

Seymour's Second Neighborhood Conjecture asserts that every oriented graph has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Combs are the graphs having no induced $C_4$, $\overline{C_4}$, $C_5$, chair or $\overline{chair}$. We characterize combs using dependency digraphs. We characterize the graphs having no induced $C_4$, $\overline{C_4}$, chair or $\overline{chair}$ using dependency digraphs. Then we prove that every oriented graph missing a comb satisfies this conjecture. We then deduce that every oriented comb and every oriented threshold graph satisfies Seymour's conjecture.

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Salman Ghazal. 2016-02-27. The Second Neighborhood Conjecture for Oriented Graphs Missing Combs. https://arxiv.org/abs/1602.08631

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