Abstract
For two integers k>0 and ell , a graph G=(V,E) is called (k,ell )-tight if |E|=k|V|-ell and i_G(X)le k|X|-ell for each Xsubseteq V for which i_G(X)ge 1, where i_G(X) denotes the number of induced edges by X. G is called (k,ell )-redundant if G-e has a spanning (k,ell )-tight subgraph for all ein E. We consider the following augmentation problem. Given a graph G=(V,E) that has a (k,ell )-tight spanning subgraph, find a graph H=(V,F) with the minimum number of edges, such that Gcup H is (k,ell )-redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is (k,ell )-tight. For general inputs, we give a polynomial algorithm when kge ell and show the NP-hardness of the problem when k<ell . Since (k,ell )-tight graphs play an important role in rigidity theory, these algorithms can be used to make several types of rigid frameworks redundantly rigid by adding a smallest set of new bars.
Highlights
Let k be a positive integer and be an integer such that < 2k
The idea of our method comes from Jackson and Jordán [10] who proved that the (k, k)-redundant edges of a (k, k)-rigid graph Gform induced subgraphs of Gwith disjoint vertex sets
First we show the NP-hardness of the following problem, called the Colored Tight Augmentation problem or CTA problem
Summary
Let k be a positive integer and be an integer such that < 2k. A (multi)graph G = (V , E) is called (k, )-sparse if iG(X ) ≤ k|X | − for all X ⊆ V for which. The idea of our method comes from Jackson and Jordán [10] who proved that the (k, k)-redundant edges of a (k, k)-rigid graph Gform induced subgraphs of Gwith disjoint vertex sets. If we contract these subgraphs into single vertices one can show that the resulting graph is (k, k)-tight for which we can use the algorithm of the reduced problem. 3 that, after using the above contraction idea for (k, )-rigid graphs with k > , the resulting graph G = (V , E ) on which we need to solve the reduced problem is (m , )-tight for some m : V → Z+ and ∈ Z for which (A0) holds. We will prove an extension of Theorem 1.1 for (m, )-tight graphs in Sect. 5 and we will give our algorithm for the reduced problem in Sect. 6 for (m, )-tight inputs
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