arXiv subjects

Hikaru Yokoi

Publications and source records attributed to Hikaru Yokoi.

2 recordsLinked to original sources

arXiv2026

Large induced subgraph with a given pathwidth in outerplanar graphs

A long-standing conjecture by Albertson and Berman in 1979 states that every planar graph of order $n$ has an induced forest with at least $\lceil \frac{n}{2} \rceil$ vertices. As a variant of this conjecture, Chappell conjectured that every planar graph of order $n$ has an induced linear forest with at least $\lceil \frac{4n}{9} \rceil$ vertices. As a partial solution to the conjecture, Pelsmajer in 2004 proved that every outerplanar graph of order $n$ has an induced linear forest with at least $\lceil \frac{4n+2}{7}\rceil$ vertices and this bound is sharp. In this paper, we investigate the order of induced subgraphs with a given pathwidth in outerplanar graphs. The above result of Pelsmajer implies that every outerplanar graph of order $n$ has an induced subgraph with pathwidth at most 1 and at least $\lceil \frac{4n+2}{7}\rceil$ vertices. We extend this to obtain a result on the maximum order of induced subgraphs with a given pathwidth in an outerplanar graph. We also give its upper bound, which generalizes Pelsmajer's construction.

cs.DM↗

arXiv2026

Structural similarity between polyhedral embeddings and their duals and its application to self-duality of pathwidth

Let $G$ be a graph embedded on a closed surface. We call $G$ a \emph{polyhedral embedding} if all facial walks are cycles, and any two of them are either disjoint or intersect in a single vertex or a single edge. In this paper, we present a new bound on the relation between the pathwidth of a polyhedral embedding and its dual. More precisely, we prove that for a polyhedral embedding $G$ on a closed surface with Euler characteristic $χ$, $\mathsf{pw}(G^*) \leq 3\ \mathsf{pw}(G)+c$, where $c$ is a constant depending only on $χ$. This result improves the coefficient of $\mathsf{pw}(G)$ in the previously known bound by Fomin and Thilikos (2007) and extends that of Amini, Huc, and Pérennes (2009) for plane graphs. Furthermore, we obtain analogous bounds on the treewidth and pathwidth of the face subdivision of a polyhedral embedding. Our approach is based on a new quantitative estimate which demonstrates the structural similarity between a polyhedral embedding and its dual.

math.CO↗