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### Algorithms for some intersection graphs [chapter]

T. Kashiwabara
1981 Lecture Notes in Computer Science
Several intersection graphs such as curves-in-the-plane graphs, circular-arc graphs, chordal graphs and interval graphs are reviewed, especially on their recognition algorithms.  ...  In this connection graph realization problem is mentioned.  ...  Trivially, any graph is an intersection graph on some appropriate model M [i] .  ...

### A proof of Pieri's formula using the generalized Schensted insertion algorithm for rc-graphs [article]

Mikhail Kogan, Abhinav Kumar
2000 arXiv   pre-print
We provide a generalization of the Schensted insertion algorithm for rc-graphs of Bergeron and Billey. The new algorithm is used to give a new proof of Pieri's formula.  ...  This algorithm will take the graph, which was rectified upto the row ℓ, and remove some intersections to produce an rc-graph of the original permutation w. Algorithm 2.  ...  Now the result of algorithm 2 will be a graph with the permutation w by Lemma 3.3, with exactly l(w) intersections. Therefore it will be an rc-graph for w.  ...

### Algorithms for clique-independent sets on subclasses of circular-arc graphs

Guillermo Durán, Min Chih Lin, Sergio Mera, Jayme Luiz Szwarcfiter
2006 Discrete Applied Mathematics
Also, we apply the algorithms to the special case of an HCA graph. The complexity of the proposed algorithm for the cardinality problem in HCA graphs is O(n).  ...  A circular-arc graph is the intersection graph of arcs on a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property.  ...  The following are some classes of graphs admitting polynomial time algorithms for the problems of determining a maximum clique-independent set: strongly chordal graphs [5, 14] ; chordal graphs with bounded  ...

### Independence and Coloring Problems on Intersection Graphs of Disks [chapter]

Thomas Erlebach, Jiří Fiala
2006 Lecture Notes in Computer Science
This chapter surveys on-line and approximation algorithms for the maximum independent set and coloring problems on intersection graphs of disks.  ...  It includes a more detailed treatment of recent upper and lower bounds on the competitive ratio of on-line algorithms for coloring such graphs.  ...  Observe that several of the presented results for disk graphs hold analogously for intersection graphs of squares and for intersection graphs of axis-aligned squares.  ...

### On cliques of Helly Circular-arc Graphs

Min Chih Lin, Ross M. McConnell, Francisco J. Soulignac, Jayme L. Szwarcfiter
2008 Electronic Notes in Discrete Mathematics
This gives a linear-time algorithm to find a proper Helly model for the clique graph of a Helly circular-arc graph.  ...  This yields the first polynomial (linear) time recognition algorithm for the clique graphs of Helly circular-arc graphs.  ...  The clique graph K(G) of G is the intersection graph of its cliques. A graph is a clique graph if it is isomorphic to K(G) for some graph G [5, 8] .  ...

### Densest k-Subgraph Approximation on Intersection Graphs [chapter]

Danny Z. Chen, Rudolf Fleischer, Jian Li
2011 Lecture Notes in Computer Science
This concept allows us to derive constant factor approximation algorithms for DS-k on many intersection graph classes, such as chordal graphs, circular-arc graphs, claw-free graphs, line graphs of -hypergraphs  ...  We study approximation solutions for the densest k-subgraph problem (DS-k) on several classes of intersection graphs.  ...  It is unlikely that there exists a PTAS for general graphs [?]. For some special graph classes and special values of k, better algorithms are known [?,?,?].  ...

### Complexity results on graphs with few cliques

Bill Rosgen, Lorna Stewart
2007 Discrete Mathematics & Theoretical Computer Science
Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a  ...  On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms.  ...  Note that polynomial time algorithms for graphs with few cliques can be made robust (that is, when given input not in the class of graphs the algorithm is designed for, the algorithm either rejects or  ...

### Efficiently Testing \$\$T\$\$ -Interval Connectivity in Dynamic Graphs [chapter]

Arnaud Casteigts, Ralf Klasing, Yessin M. Neggaz, Joseph G. Peters
2015 Lecture Notes in Computer Science
We show that Ω(δ) such operations are required to solve both problems, and we present optimal O(δ) online algorithms for both problems.  ...  We assume that the changes between two consecutive graphs are arbitrary, and that two operations, binary intersection and connectivity testing, are available to solve the problems.  ...  . , G ℓ [δ − ℓ + 1], for some k + 1 ≤ ℓ ≤ 2k.  ...

### Efficiently Testing T-Interval Connectivity in Dynamic Graphs [article]

Arnaud Casteigts, Ralf Klasing, Yessin M. Neggaz, Joseph G. Peters
2017 arXiv   pre-print
We show that Ω(δ) such operations are required to solve both problems, and we present optimal O(δ) online algorithms for both problems.  ...  We assume that the changes between two consecutive graphs are arbitrary, and that two operations, binary intersection and connectivity testing, are available to solve the problems.  ...  . , G ℓ [δ − ℓ + 1], for some k + 1 ≤ ℓ ≤ 2k.  ...

### The graph isomorphism problem on geometric graphs

Ryuhei Uehara
2014 Discrete Mathematics & Theoretical Computer Science
We also show that the GI problem is as hard as the problem on general graphs even for grid unit intersection graphs on a torus, that partially solves an open problem.  ...  Sometimes the GI problem becomes polynomial time solvable when we add some restrictions on some graph classes.  ...  Since the Reingold's log-space algorithm for undirected connectivity (Reingold (2008) ), some log-space algorithms for planar graphs (Datta et al. (2009) ) and interval graphs (Köbler et al. (2011)  ...

### An E log E Line Crossing Algorithm for Levelled Graphs [chapter]

1999 Lecture Notes in Computer Science
This paper describes an algorithm for leveled graphs, based on the classification of edges that is O(e log e) where e is the number of edges.  ...  It is also a performance bottleneck for Sugiyama-style layout algorithms.  ...  We have discovered by experience that counting line intersections is a performance bottleneck for the entire algorithm for large graphs.  ...

### The topological drawing of a graph: Construction methods

S. V. Kurapov, A. V. Tolok
2013 Automation and remote control
This paper considers construction algorithms for the topological 2-D drawing of a graph. These algorithms allow to store, describe and modify the existing information on the drawing of a graph.  ...  Finally, we introduce necessary notions and structures to solve the problem of the topological 2-D drawing of a graph.  ...  Lee's algorithm belongs to metrical methods, since connections are considered in some subspace of R 2 with the Euclidean metric.  ...

### Optimization problems in multiple subtree graphs

Danny Hermelin, Dror Rawitz
2011 Discrete Applied Mathematics
We consider various optimization problems in t-subtree graphs, the intersection graphs of t-subtrees, where a t-subtree is the union of t disjoint subtrees of some tree.  ...  We present approximation algorithms for the Maximum Independent Set, Minimum Coloring, Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique problems in t-subtree graphs.  ...  graphs by Martin), but also for the numerous hallway discussions with him about life and academia.  ...

### Optimization Problems in Multiple Subtree Graphs [chapter]

Danny Hermelin, Dror Rawitz
2010 Lecture Notes in Computer Science
We consider various optimization problems in t-subtree graphs, the intersection graphs of t-subtrees, where a t-subtree is the union of t disjoint subtrees of some tree.  ...  We present approximation algorithms for the Maximum Independent Set, Minimum Coloring, Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique problems in t-subtree graphs.  ...  graphs by Martin), but also for the numerous hallway discussions with him about life and academia.  ...

### The clique-separator graph for chordal graphs

Louis Ibarra
2009 Discrete Applied Mathematics
We present an algorithm that constructs the clique-separator graph of a chordal graph in O(n 3 ) time and of an interval graph in O(n 2 ) time, where n is the number of vertices in the graph.  ...  We present structural properties of the clique-separator graph and additional properties when the chordal graph is an interval graph, proper interval graph, or split graph.  ...  Then Intersect(N) ⊆ Intersect(S) and we must show that Intersect(N) ⊂ Intersect(S). Since S N, it follows that N is in some connected component of G − P reds(S).  ...
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