Sparse graphs and an augmentation problem

<jats:p>For two integers <jats:inline-formula><jats:alternatives><jats:tex-math>$$k>0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\ell $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>, a graph <jats:inline-formula><jats:alternatives><jats:tex-math>$$G=(V,E)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> is called <jats:inline-formula><jats:alternatives><jats:tex-math>$$(k,\ell )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>-tight if <jats:inline-formula><jats:alternatives><jats:tex-math>$$|E|=k|V|-\ell $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>E</mml:mi> <mml:mo>|</mml:mo> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mo>|</mml:mo> <mml:mi>V</mml:mi> <mml:mo>|</mml:mo> <mml:mo>-</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$i_G(X)\le k|X|-\ell $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>X</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>-</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> for each <jats:inline-formula><jats:alternatives><jats:tex-math>$$X\subseteq V$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> for which <jats:inline-formula><jats:alternatives><jats:tex-math>$$i_G(X)\ge 1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$i_G(X)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> denotes the number of induced edges by <jats:italic>X</jats:italic>. <jats:italic>G</jats:italic> is called <jats:inline-formula><jats:alternatives><jats:tex-math>$$(k,\ell )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>-redundant if <jats:inline-formula><jats:alternatives><jats:tex-math>$$G-e$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>-</mml:mo> <mml:mi>e</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> has a spanning <jats:inline-formula><jats:alternatives><jats:tex-math>$$(k,\ell )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>-tight subgraph for all <jats:inline-formula><jats:alternatives><jats:tex-math>$$e\in E$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>E</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. We consider the following augmentation problem. Given a graph <jats:inline-formula><jats:alternatives><jats:tex-math>$$G=(V,E)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> that has a <jats:inline-formula><jats:alternatives><jats:tex-math>$$(k,\ell )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>-tight spanning subgraph, find a graph <jats:inline-formula><jats:alternatives><jats:tex-math>$$H=(V,F)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> with the minimum number of edges, such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$G\cup H$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>∪</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> is <jats:inline-formula><jats:alternatives><jats:tex-math>$$(k,\ell )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>-redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is <jats:inline-formula><jats:alternatives><jats:tex-math>$$(k,\ell )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>-tight. For general inputs, we give a polynomial algorithm when <jats:inline-formula><jats:alternatives><jats:tex-math>$$k\ge \ell $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mm

[1]  Tibor Jordán,et al.  Algorithms for Graph Rigidity and Scene Analysis , 2003, ESA.

[2]  Tibor Jordán,et al.  Minimum Cost Globally Rigid Subgraphs , 2019, Bolyai Society Mathematical Studies.

[3]  Tibor Jordán,et al.  Generic global rigidity of body-hinge frameworks , 2016, J. Comb. Theory, Ser. B.

[4]  Bill Jackson,et al.  Brick partitions of graphs , 2010, Discret. Math..

[5]  Tibor Jordán,et al.  II – Combinatorial Rigidity: Graphs and Matroids in the Theory of Rigid Frameworks , 2016 .

[6]  Audrey Lee-St. John,et al.  Pebble game algorithms and sparse graphs , 2007, Discret. Math..

[7]  H. Pollaczek-Geiringer.  Über die Gliederung ebener Fachwerke , 1927 .

[8]  W. Whiteley.  Rigidity of Molecular Structures: Generic and Geometric Analysis , 2002 .

[9]  Bill Jackson,et al.  Global rigidity of generic frameworks on the cylinder , 2016, J. Comb. Theory, Ser. B.

[10]  Alfredo García Olaverri,et al.  Augmenting the Rigidity of a Graph in R2 , 2011, Algorithmica.

[11]  B. Anderson,et al.  Development of redundant rigidity theory for formation control , 2009 .

[12]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[13]  Tiong-Seng Tay,et al.  Henneberg's Method for Bar and Body Frameworks , 1991 .

[14]  Brian D. O. Anderson,et al.  A Theory of Network Localization , 2006, IEEE Transactions on Mobile Computing.

[15]  Tibor Jordán,et al.  Generic global rigidity of body-bar frameworks , 2013, J. Comb. Theory B.

[16]  C. Nash-Williams.  Decomposition of Finite Graphs Into Forests , 1964 .

[17]  A. Frank.  Connections in Combinatorial Optimization , 2011 .

[18]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[19]  Anthony Nixon,et al.  Rigidity of Frameworks Supported on Surfaces , 2010, SIAM J. Discret. Math..

[20]  Robert E. Tarjan,et al.  Augmentation Problems , 1976, SIAM J. Comput..

[21]  B. Hendrickson,et al.  An Algorithm for Two-Dimensional Rigidity Percolation , 1997 .

[22]  Jacobs,et al.  Generic rigidity percolation: The pebble game. , 1995, Physical review letters.

[23]  András Frank,et al.  Combined Connectivity Augmentation and Orientation Problems , 2001, IPCO.

[24]  Walter Whiteley,et al.  Some matroids from discrete applied geometry , 1996 .

[25]  Ileana Streinu,et al.  Sparse Hypergraphs and Pebble Game Algorithms , 2007, Eur. J. Comb..

[26]  M. Lorea,et al.  On matroidal families , 1979, Discret. Math..

[27]  Zsolt Fekete,et al.  Egerváry Research Group on Combinatorial Optimization Uniquely Localizable Networks with Few Anchors , 2022 .

[28]  Bruce Hendrickson,et al.  Conditions for Unique Graph Realizations , 1992, SIAM J. Comput..