On bounds on bend number of split and cocomparability graphs

A path is a simple, piecewise linear curve made up of alternating horizontal and vertical line segments in the plane. A $k$-bend path is a path made up of at most $k + 1$ line segments. A $B_k$-VPG representation of a graph is a collection of $k$-bend paths such that each path in the collection represents a vertex of the graph and two such paths intersect if and only if the vertices they represent are adjacent in the graph. The graphs that have a $B_k$-VPG representation are called $B_k$-VPG graphs. It is known that the poset dimension $dim(G)$ of a cocomparability graph $G$ is greater than or equal to its bend number $bend(G)$. Cohen et al. ({\textsc{order 2015}}) asked for examples of cocomparability graphs with low bend number and high poset dimension. We answer this question by proving that for each $m, t \in \mathbb{N}$, there exists a cocomparability graph $G_{t,m}$ with $t < bend(G_{t,m}) \leq 4t+29$ and $dim(G_{t,m})-bend(G_{t,m})>m$. Techniques used to prove the above result, allows us to partially address the open question posed by Chaplick et al. ({\textsc{wg 2012}}) who asked whether $B_k$-VPG-chordal $\subsetneq$ $B_{k+1}$-VPG-chordal for all $k \in \mathbb{N}$. We address this by proving that there are infinitely many $m \in \mathbb{N}$ such that $B_m$-VPG-split $\subsetneq$ $B_{m+1}$-VPG-split which provides infinitely many positive examples. We use the same techniques to prove that, for all $t \in \mathbb{N}$, $B_t$-VPG-$Forb(C_{\geq 5})$ $\subsetneq$ $B_{4t+29}$-VPG-$Forb(C_{\geq 5})$, where $Forb(C_{\geq 5})$ denotes the family of graphs that does not contain induced cycles of length greater than 4. Furthermore, we show that for all $t \in \mathbb{N}$, $PB_t$-VPG-split $\subsetneq PB_{36t+80}$-VPG-split, where $PB_t$-VPG denotes the class of graphs with proper bend number at most $t$.