On bounds on bend number of classes of split and cocomparability graphs
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A k-bend path is a rectilinear curve made up of 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. In this paper, we address the open question posed by Chaplick et al. (wg 2012) asking whether B_k-VPG-chordal ⊊ B_k+1-VPG-chordal for all k ∈N, where B_k-VPG denotes the class of graphs with bend number at most k. We prove there are infinitely many m ∈N such that B_m-VPG-split ⊊ B_m+1-VPG-split which provides infinitely many positive examples with respect to the open question. We also 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 PB_t-VPG-split ⊊ PB_36t+80-VPG-split. for all t ∈N, where PB_t-VPG denotes the class of graphs with proper bend number at most t. In this paper, we study the relationship between poset dimension and its bend number. 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 the above open 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) is greater than m.
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