The relationship between the thresh old dimension of split graphs and various dimensional parameters
ARIDAM III Selected papers on Third advanced research institute of discrete applied mathematics
A special planar satisfiability problem and a consequence of its NP-completeness
Discrete Applied Mathematics
A constant factor approximation algorithm for boxicity of circular arc graphs
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Polynomial time and parameterized approximation algorithms for boxicity
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
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A k-dimensional box is the Cartesian product R"1xR"2x...xR"k where each R"i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R"1xR"2x...xR"k where each R"i is a closed interval on the real line of the form [a"i,a"i+1]. The cubicity of G, denoted as cub(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. The threshold dimension of a graph G(V,E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on n vertices with a factor of O(n^0^.^5^-^@e) for any @e0 unless NP=ZPP. From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on n vertices with factor O(n^0^.^5^-^@e) for any @e0 unless NP=ZPP. In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is NP-complete to determine whether a given split graph has boxicity at most 3.