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Recognizing Helly Edge-Path-Tree graphs and their clique graphs

Nicola Apollonio, Massimiliano Caramia
2011 Discrete Applied Mathematics  
We present a unifying procedure for recognizing intersection graphs of Helly families of paths in a tree and their clique graphs.  ...  The Helly property makes it possible to look at these recognition problems as variants of the Graph Realization Problem, namely, the problem of recognizing Edge-Path-Tree matrices.  ...  the class of clique-graphs of class G. A graph is clique-Helly if K(G) is Helly. It is hereditary clique-Helly if each induced subgraph of G is clique-Helly.  ... 
doi:10.1016/j.dam.2011.02.008 fatcat:crnsduwvq5fvnf3xqns3un4s3a

Faster recognition of clique-Helly and hereditary clique-Helly graphs

Min Chih Lin, Jayme L. Szwarcfiter
2007 Information Processing Letters  
The best algorithms currently known to recognize clique-Helly and hereditary clique-Helly graphs have complexities O(nm 2 ) and O(n 2 m), respectively for a graph with n vertices and m edges.  ...  A graph G is clique-Helly when the family of its (maximal) cliques is Helly, while G is hereditary clique-Helly when every induced subgraph of it is clique-Helly.  ...  Currently known algorithms recognize clique-Helly graphs in O(nm 2 ) time, and hereditary clique-Helly in O(n 2 m) time.  ... 
doi:10.1016/j.ipl.2007.02.017 fatcat:437fimvx4rctljtdeh6mtxgbie

Computational aspects of the Helly property: a survey

Mitre C. Dourado, Fabio Protti, Jayme L. Szwarcfiter
2006 Journal of the Brazilian Computer Society  
In 1923, Eduard Helly published his celebrated theorem, which originated the well known Helly property.  ...  In this work, we survey computational aspects of the Helly property. The main focus is algorithmic.  ...  Theorem 3.5 [28] CLIQUE-HELLY SANDWICH GRAPH is NP-complete. Disk-Helly graphs Disk-Helly graphs can also be recognized in polynomial time.  ... 
doi:10.1590/s0104-65002006000200002 fatcat:xoccurn6tnctpdyxuphsvnhsgy

Computational aspects of the Helly property: a survey

Mitre C. Dourado, Fábio Protti, Jayme L. Szwarcfiter
2006 Journal of the Brazilian Computer Society  
In 1923, Eduard Helly published his celebrated theorem, which originated the well known Helly property.  ...  In this work, we survey computational aspects of the Helly property. The main focus is algorithmic.  ...  Theorem 3.5 [28] CLIQUE-HELLY SANDWICH GRAPH is NP-complete. Disk-Helly graphs Disk-Helly graphs can also be recognized in polynomial time.  ... 
doi:10.1007/bf03192385 fatcat:g576bhjqfbbdhhbqxgaodx2aiy

Complexity Aspects of the Helly Property: Graphs and Hypergraphs

Mitre C. Dourado, Fábio Protti, Jayme L. Szwarcfiter
2009 Electronic Journal of Combinatorics  
In this work, we survey complexity aspects of the Helly property. The main focus is on characterizations of several classes of graphs and hypergraphs related to the Helly property.  ...  In 1923, Eduard Helly published his celebrated theorem, which originated the well known Helly property.  ...  Hence, G is a (p, q)-clique-Helly graph. From the above theorem one can recognize (p, q)-clique-Helly graphs in polynomial time if p and q are fixed.  ... 
doi:10.37236/38 fatcat:kqezaclnwbddraypsil6n5ltry

On the strong p-Helly property

Mitre C. Dourado, Fábio Protti, Jayme L. Szwarcfiter
2008 Discrete Applied Mathematics  
Further, we apply the concept of strong p-Helly hypergraphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs.  ...  For p = 2, this class is equivalent to that of hereditary clique-Helly graphs [E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220].  ...  Similarly, the algorithm for recognizing p-clique-Helly graphs is not suitable for recognizing hereditary p-clique-Helly graphs either, because the number of induced subgraphs may also be exponential in  ... 
doi:10.1016/j.dam.2007.05.047 fatcat:ly2qavmkpjclfhb6jki5hi3zda

Characterization and recognition of Helly circular-arc clique-perfect graphs

Flavia Bonomo, Guillermo Durán
2005 Electronic Notes in Discrete Mathematics  
In this work we characterize clique-perfect graphs by a restricted list of minimal forbidden induced subgraphs when the graph is a Helly circular-arc graph.  ...  A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques.  ...  A graph is clique-Helly (CH) if its cliques satisfy the Helly property, and it is hereditary clique-Helly (HCH) if H is clique-Helly for every induced subgraph H of G.  ... 
doi:10.1016/j.endm.2005.06.026 fatcat:2wkk25kfp5a6zlmf7pgqkw5ddi

Recognizing clique graphs of directed and rooted path graphs

Erich Prisner, Jayme L. Szwarcfiter
1999 Discrete Applied Mathematics  
We describe characterizations for the classes of clique graphs of directed and rooted path graphs.  ...  The characterizations lead to polynomial time algorithms for recognizing graphs of these classes. 0 1999 Elsevier Science B.  ...  Recognizing if G is a clique-Helly graph can be done in 0(n3m) steps [17]. Constructing G' from G requires O(n) steps.  ... 
doi:10.1016/s0166-218x(99)00028-1 fatcat:uznrmhjw5zbqfogmnpf3m444q4

The Complexity of Helly-$B_{1}$ EPG Graph Recognition

Claudson F. Bornstein, Martin Charles Golumbic, Tanilson D. Santos, Uéverton S. Souza, Jayme L. Szwarcfiter
2019 Discrete Mathematics & Theoretical Computer Science  
Moreover, we show that the problem of recognizing Helly-$B_1$-EPG graphs is NP-complete, and it remains NP-complete even when restricted to 2-apex and 3-degenerate graphs.  ...  In this paper, we show that given a graph $G$ and an integer $k$, the problem of determining whether $G$ admits a $B_k$-EPG representation whose edge-intersections of paths satisfy the Helly property,  ...  Regarding the complexity of recognizing B k -EPG graphs, only the complexity of recognizing a few of these sub-classes of EPG graphs has been determined: B 0 -EPG graphs can be recognized in polynomial  ... 
doi:10.23638/dmtcs-22-1-19 fatcat:estm7plbifdxthqvvtbpq2zwly

The Complexity of Helly-B_1 EPG Graph Recognition [article]

Claudson F. Bornstein and Martin Charles Golumbic and Tanilson D. Santos and Uéverton S. Souza and Jayme L. Szwarcfiter
2020 arXiv   pre-print
Moreover, we show that the problem of recognizing Helly-B_1-EPG graphs is NP-complete, and it remains NP-complete even when restricted to 2-apex and 3-degenerate graphs.  ...  In this paper, we show that given a graph G and an integer k, the problem of determining whether G admits a B_k-EPG representation whose edge-intersections of paths satisfy the Helly property, so-called  ...  Regarding the complexity of recognizing B k -EPG graphs, only the complexity of recognizing a few of these sub-classes of EPG graphs has been determined: B 0 -EPG graphs can be recognized in polynomial  ... 
arXiv:1906.11185v3 fatcat:4ucb4lehnnc7po3ciuxjenj22u

Helly property, clique raphs, complementary graph classes, and sandwich problems

Mitre C. Dourado, Priscila Petito, Rafael B. Teixeira, Celina M. H. de Figueiredo
2008 Journal of the Brazilian Computer Society  
A graph is clique-Helly when its family of cliques satisfies the Helly property. A graph is hereditary clique-Helly when all of its induced subgraphs are clique-Helly.  ...  We show that the sandwich problems for the graph classes: clique, clique-Helly, hereditary clique-Helly, and clique-Helly nonhereditary are all NP -complete.  ...  Clearly, from Theorems 2 and 3 one can also recognize in polynomial time whether a graph is clique-Helly nonhereditary.  ... 
doi:10.1007/bf03192558 fatcat:imrynj3dgzbihpe45tl5xn23hi

Helly property, clique graphs, complementary graph classes, and sandwich problems

Mitre C. Dourado, Priscila Petito, Rafael B. Teixeira, Celina M. H. de Figueiredo
2008 Journal of the Brazilian Computer Society  
A graph is clique-Helly when its family of cliques satisfies the Helly property. A graph is hereditary clique-Helly when all of its induced subgraphs are clique-Helly.  ...  We show that the sandwich problems for the graph classes: clique, clique-Helly, hereditary clique-Helly, and clique-Helly nonhereditary are all NP -complete.  ...  Clearly, from Theorems 2 and 3 one can also recognize in polynomial time whether a graph is clique-Helly nonhereditary.  ... 
doi:10.1590/s0104-65002008000200004 fatcat:tzu2sfmskjdblpg4gqfs66xto4

On the clique operator [chapter]

Marisa Gutierrez, JoÃo Meidanis
1998 Lecture Notes in Computer Science  
Among all the better studied graph operators, K seems to be the richest one and many questions regarding it remain open. In particular, it is not known whether recognizing a clique graph is in P.  ...  The clique operator K maps a graph G into its clique graph, which is the intersection graph of the (maximal) cliques of G.  ...  A graph is HeUy when the family of its cliques is Helly. We denote by 74 the class of Helly graphs. A family ~r is conformal when the cliques of Jr2 are all members of ~'.  ... 
doi:10.1007/bfb0054327 fatcat:racvmnrqpzhddbnp6y74fpnyua

On the generalized Helly property of hypergraphs, cliques, and bicliques [article]

Mitre C. Dourado, Luciano N. Grippo, Martín D. Safe
2022 arXiv   pre-print
In addition, we generalize to (p,q)-clique-Helly graphs the characterization of p-clique-Helly graphs in terms of expansions and give different characterizations of hereditary (p,q)-clique-Helly graphs  ...  A graph is (p,q)-clique-Helly if the family of its maximal cliques has the (p,q)-the Helly property and hereditary (p,q)-clique-Helly if each of its induced subgraphs is (p,q)-clique-Helly.  ...  The (p, q)-clique-Helly property of graphs In this section, we study the problems of characterizing and recognizing (p, q)-clique-Helly graphs and hereditary (p, q)-clique-Helly graphs.  ... 
arXiv:2201.12610v1 fatcat:auwnntvpinaobjzsq2j24h74yi

Recognizing clique graphs of directed edge path graphs

Marisa Gutierrez, João Meidanis
2003 Discrete Applied Mathematics  
In this work, we show that the clique graphs of these graphs are exactly the two sections of the same kind of path families, and give a polynomial time recognition algorithm for them. ?  ...  Directed edge path graphs are the intersection graphs of directed paths in a directed tree, viewed as sets of edges. They were studied by Monma and Wei (J. Comb.  ...  Since recognizing clique-Helly graphs and DE graphs can be done in polynomial time [7, 9] , and the number of cliques of a duallyDE graph is also polynomial by Theorem 5, the entire procedure takes polynomial  ... 
doi:10.1016/s0166-218x(02)00203-2 fatcat:x2navd3fdvhlzcrm2nmokmnqte
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