Boxicity of Permutation Graphs
نویسندگان
چکیده
An axis parallel d-dimensional box is the cartesian product R1 × R2 × · · · × Rd where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer d such that G can be represented as the intersection graph of a collection of d-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. Permutation graphs form a well-known subclass of perfect graphs. A permutation graph is a graph that can be represented as the intersection graph of a family of line segments that connect two parallel lines in the Euclidean plane. Let G be a permutation graph with chromatic number χ(G) and maximum clique size ω(G). We will show that box(G) ≤ χ(G) = ω(G) and this bound is tight.
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