Abstract
For two integers \(k>0\) and \(\ell \), a graph \(G=(V,E)\) is called \((k,\ell )\)-tight if \(|E|=k|V|-\ell \) and \(|E(X)|\le k|X|-\ell \) for all \(X\subseteq V\) for which \(k|X|-\ell \ge 0\). G is called \((k,\ell )\)-redundant if \(G-e\) has a spanning \((k,\ell )\)-tight subgraph for all \(e\in E\). 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 minimum number of edges, such that \(G+H\) is \((k,\ell )\)-redundant.
In this paper, 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 \(k\ge \ell \) 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.
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Acknowledgements
Project no. NKFI-128673 has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the FK_18 funding scheme. The first author was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the ÚNKP-19-4 New National Excellence Program of the Ministry for Innovation and Technology. The authors are grateful to Tibor Jordán for the inspiring discussions and his comments.
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Appendices
Appendix A: Sketch of the Proof of Lemma 7
Here we give the main steps of the proof of Lemma 7. Recall that we assume (A) and \(\mathcal {C}\) is the family of all \({(m,\ell )}\)-MCT sets of G. When \(\ell \le 0\), the proof is straightforward from Lemmas 1 and 6. In the case of \(\ell >0\), we need to use our assumptions (A1), (A2), and (A3) at some points. Our first statement follows by Lemma 1.
Lemma 13
If X and Y are two \({(m,\ell )}\)-MCT sets in G, such that \(X \cap Y \ne \emptyset \), then \({m}(V-(X \cup Y)) <\ell \). In particular, \(|X \cup Y| \ge |V|-1\).
If \(X\cup Y =V\) holds whenever X and Y are intersecting MCT sets, then the proof is straightforward. Hence we may assume that \(|X\cup Y| = |V|-1\) for some \(X,Y \subset V\). For a vertex \(v\in V\), let \(\varvec{\mathcal {C}(v)\,}:=\{C\in \mathcal {C}:v\notin C\}\). The first key of the proof is the following lemma which can be proved by Lemmas 6 and 13. Assumption (A1) is used in its proof.
Lemma 14
Suppose that \(\ell >0\). Assume that there exists two \({(m,\ell )}\)-MCT sets \(X,Y\in \mathcal {C}\) such that \(X\cap Y\ne \emptyset \) and \(X\cup Y=V-v\) for some \(v\in V\). Then \(\mathcal {C}(v)\) is a co-partition of \(V-v\) with \(|\mathcal {C}(v)|\ge 3\) or there exists a vertex \(u\in V-v\) such that \(T(uv)=G\).
For a vertex \(v\in V\) and a set \(W\subseteq V-v\), let \(\varvec{\widetilde{W}^v\,}:={V-v-W}\). Lemma 6 implies that, if we take two members \(W_1\) and \(W_2\) of the co-partition \(\mathcal {C}(v)\) and take \(w_1\in \widetilde{W}_1^v\) and \(w_2 \in \widetilde{W}_2^v\), then V is the only \({(m,\ell )}\)-tight set in G which contains \(w_1,w_2\) and v. Using this observation we can prove a much stronger statement which claims that in many cases \(V(w_1w_2)\) is also V.
Lemma 15
Suppose that \(\ell >0\). Let \(v\in V\) be a vertex for which the family \(\mathcal {C}(v)\) is a co-partition of \(V-v\) with \(|\mathcal {C}(v)|\ge 3\). Suppose that there exists a vertex \(u\in V-v\) with \(m(u)\le m(v)\). Let \(W_1,W_2\in \mathcal {C}(v)\) and let \(w_1\in \widetilde{W}^v_1\) and \(w_2 \in \widetilde{W}^v_2\). Suppose that \(V'\) is an \({(m,\ell )}\)-tight set in G with \(w_1,w_2 \in V'\). Then either \(V'=V\) or \(V'=\{w_1,w_2\}\). In particular, either \(V(w_1w_2)=V\) (and \(T(w_1w_2)=G\)) or \(V(w_1w_2)=\{w_1,w_2\}\).
Based on Lemma 15, using assumption (A2) one can prove the following.
Lemma 16
Suppose that \(\ell >0\). Let \(v\in V\) be a vertex for which the family \(\mathcal {C}(v)\) is a co-partition of \(V-v\) with \(|\mathcal {C}(v)|\ge 3\). Then \(m(v)< m(u)\) holds for every \(u\in V-v\) or there exist two vertices \(x,y\in V-v\) such that \(T(xy)=G\).
Finally, to finish the proof of Lemma 7, we can assume by Lemma 16 that, whenever X and Y are intersecting MCT sets with \(X\cup Y=V-v\) for a \(v \in V\), then \(m(v)< m(u)\) holds for every \(u\in V-v\). In this case, one can prove that \(\{v\}\) is an MCT set and hence \(d(v)=m(v)\). Now the proof follows from (A0), (A3) and the fact that the degree of v in any \({(m,\ell )}\)-tight subgraph on more than 3 vertices is at least m(v).
Appendix B: The algorithm of Lemma 12
In this section we give the algorithm of Lemma 12. First we solve the case when we have an MCT set consisting of a single vertex.
Lemma 17
Assume (A). If we are given an \({(m,\ell )}\)-MCT singleton set \(C=\{v\}\), then we can check if there exists an edge e such that \(T(e)=G\) in \(O(|V|^2)\) time.
Based on the steps of the proof of Lemma 7 in Appendix A we can provide the following algorithm for Lemma 12.
Algorithm 4
Proof Sketch of the Correctness of Algorithm 4: By Lemma 17 and by the correctness of Algorithm 1, we can see that any output from Steps \(2-9\) is correct. It is also clear by Algorithm 1 that the sets Z, C and S are MCT sets, thus the output in Step 10 is also correct by Lemma 7. Hence we only need to see that Step 11 gives a suitable edge. By Lemma 14, if no suitable edge is given, then \(\mathcal {C}(v)\), \(\mathcal {C}(z)\) and \(\mathcal {C}(c)\) are co-partitions. However, this contradicts Lemma 15. \(\square \)
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Király, C., Mihálykó, A. (2020). Sparse Graphs and an Augmentation Problem. In: Bienstock, D., Zambelli, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2020. Lecture Notes in Computer Science(), vol 12125. Springer, Cham. https://doi.org/10.1007/978-3-030-45771-6_19
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