On bounds on bend number of split and cocomparability graphs
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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. (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 ∈N, there exists a cocomparability graph G_t,m with t < bend(G_t,m) ≤ 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. (wg 2012) who asked whether B_k-VPG-chordal ⊊ B_k+1-VPG-chordal for all k ∈N. We address this by proving that there are infinitely many m ∈N such that B_m-VPG-split ⊊ B_m+1-VPG-split which provides infinitely many positive examples. We use the same techniques to prove that, for all t ∈N, B_t-VPG-Forb(C_≥ 5) ⊊ B_4t+29-VPG-Forb(C_≥ 5), where Forb(C_≥ 5) denotes the family of graphs that does not contain induced cycles of length greater than 4. Furthermore, we show that for all t ∈N, PB_t-VPG-split ⊊ PB_36t+80-VPG-split, where PB_t-VPG denotes the class of graphs with proper bend number at most t.
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