Deepak Rajendraprasad

Separation Dimension of Graphs and Hypergraphs

Separation dimension of a hypergraph H, denoted by

Rainbow colouring of split graphs

\pi (H)

Heterochromatic paths in edge colored graphs without small cycles and heterochromatic-triangle-free graphs

π(H), is the smallest natural number k so that the vertices of H can be embedded in

The Induced Separation Dimension of a Graph

\mathbb {R}^k

Separation dimension and sparsity

Rk such that any two disjoint edges of H can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph H is equal to the boxicity of the line graph of H. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension. In this paper, we study the separation dimension of hypergraphs and graphs.

Edge-intersection graphs of boundary-generated paths in a grid

In this paper, we study testing of sequence properties that are defined by forbidden order patterns. A sequence f : {1, . . . , n} → ℝ of length n contains a pattern π ∈ 𝔖k (𝔖k is the group of permutations of k elements), iff there are indices i1 f(iy) whenever π(x) > π(y). If f does not contain π, we say f is π-free. For example, for π = (2, 1), the property of being π-free is equivalent to being non-decreasing, i.e. monotone. The property of being (k, k − 1, . . . , 1)-free is equivalent to the property of having a partition into at most k − 1 non-decreasing subsequences. Let π ∈ 𝔖k, k constant, be a (forbidden) pattern. Assuming f is stored in an array, we consider the property testing problem of distinguishing the case that f is π-free from the case that f differs in more than ϵn places from any π-free sequence. We show the following results: There is a clear dichotomy between the monotone patterns and the non-monotone ones: • For monotone patterns of length k, i.e., (k, k − 1, . . . , 1) and (1, 2, . . . , k), we design non-adaptive one-sided error ϵ-tests of (ϵ−1 log n)O(k2) query complexity. • For non-monotone patterns, we show that for any size-k non-monotone π, any non-adaptive one-sided error ϵ-test requires at least [EQUATION] queries. This general lower bound can be further strengthened for specific non-monotone k-length patterns to Ω(n1−2/(k+1)). On the other hand, there always exists a non-adaptive one-sided error ϵ-test for π ∈ 𝔖k with O(ϵ−1/kn1−1/k) query complexity. Again, this general upper bound can be further strengthened for specific non-monotone patterns. E.g., for π = (1, 3, 2), we describe an ϵ-test with (almost tight) query complexity of [EQUATION]. Finally, we show that adaptivity can make a big difference in testing non-monotone patterns, and develop an adaptive algorithm that for any π ∈ 𝔖3, tests π-freeness by making (ϵ−1 log n)O(1) queries. For all algorithms presented here, the running times are linear in their query complexity.