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6 results on '"Santos, Tanilson D."'

1. Edge Intersection Graphs of Paths on a Triangular Grid

Author

de Luca, Vitor T. F., Mazzoleni, María Pía, Oliveira, Fabiano S., Santos, Tanilson D., and Szwarcfiter, Jayme L.

Subjects
Computer Science - Discrete Mathematics, Computer Science - Data Structures and Algorithms, 05C85, 05C05
Abstract

We introduce a new class of intersection graphs, the edge intersection graphs of paths on a triangular grid, called EPGt graphs. We show similarities and differences from this new class to the well-known class of EPG graphs. A turn of a path at a grid point is called a bend. An EPGt representation in which every path has at most $k$ bends is called a B$_k$-EPGt representation and the corresponding graphs are called B$_k$-EPGt graphs. We provide examples of B$_{2}$-EPG graphs that are B$_{1}$-EPGt. We characterize the representation of cliques with three vertices and chordless 4-cycles in B$_{1}$-EPGt representations. We also prove that B$_{1}$-EPGt graphs have Strong Helly number $3$. Furthermore, we prove that B$_{1}$-EPGt graphs are $7$-clique colorable., Comment: 19 pages, 12 figures

Published
2022

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

Author

Bornstein, Claudson F., Golumbic, Martin Charles, Santos, Tanilson D., Souza, Uéverton S., and Szwarcfiter, Jayme L.

Subjects
Computer Science - Discrete Mathematics, Computer Science - Computational Complexity, Computer Science - Data Structures and Algorithms
Abstract

Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the intersection graph class of edge paths on a grid. An EPG graph $G$ is a graph that admits a representation where its vertices correspond to paths in a grid $Q$, such that two vertices of $G$ are adjacent if and only if their corresponding paths in $Q$ have a common edge. If the paths in the representation have at most $k$ bends, we say that it is a $B_k$-EPG representation. A collection $C$ of sets satisfies the Helly property when every sub-collection of $C$ that is pairwise intersecting has at least one common element. 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 Helly-$B_k$-EPG representation, is in NP, for every $k$ bounded by a polynomial function of $|V(G)|$. 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.

Published
2019
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4. The Complexity of Helly-$B_{1}$ EPG Graph Recognition

Author

Bornstein, Claudson F., Golumbic, Martin Charles, Santos, Tanilson D., Souza, Uéverton S., and Szwarcfiter, Jayme L.

Subjects
FOS: Computer and information sciences, Computer Science - Computational Complexity, Discrete Mathematics (cs.DM), Computer Science - Data Structures and Algorithms, Data Structures and Algorithms (cs.DS), Computational Complexity (cs.CC), Computer Science - Discrete Mathematics
Abstract

Discrete Mathematics & Theoretical Computer Science ; vol. 22 no. 1 ; Graph Theory ; 1365-8050, Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the intersection graph class of edge paths on a grid. An EPG graph $G$ is a graph that admits a representation where its vertices correspond to paths in a grid $Q$, such that two vertices of $G$ are adjacent if and only if their corresponding paths in $Q$ have a common edge. If the paths in the representation have at most $k$ bends, we say that it is a $B_k$-EPG representation. A collection $C$ of sets satisfies the Helly property when every sub-collection of $C$ that is pairwise intersecting has at least one common element. 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 Helly-$B_k$-EPG representation, is in NP, for every $k$ bounded by a polynomial function of $|V(G)|$. 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.

Published
2020
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5. HELLY AND STRONG HELLY NUMBERS OF Bk-EPG AND Bk-VPG GRAPHS.

Author

BORNSTEIN, CLAUDSON F., MORGENSTERN, GILA, SANTOS, TANILSON D., SOUZA, UÉVERTON S., and SZWARCFITER, JAYME L.

Subjects
*ORTHOGONALIZATION, *PROBLEM solving, *INTEGERS
Abstract

EPG graphs were introduced by Golumbic, Lypshteyn, and Stern in 2009 and consist of the intersection graphs of sets of paths on the orthogonal grid, whose intersections are taken considering the edges of the paths. If the intersections of the paths consider the vertices and not the edges, the resulting graph class is called VPG graphs. A path P is a Bk-path if it contains at most k bends. Bk-EPG and Bk-VPG graphs are the intersection graphs of Bk-paths on the orthogonal grid, considering the intersection of edges and vertices, respectively. A family F is h-Helly when every h-intersecting sub-family F' of it satisfies core(F') ≠ 0. If for every subfamily F' of F, there are h subsets whose core equals the core of F', then F is said to be strong h-Helly. The Helly number of the family F is the least integer h, such that F is h-Helly. Similarly, the strong Helly number of F is the least h, for which F is strong h-Helly. In this paper, we solve the problem of determining both the Helly and strong Helly numbers, for Bk-EPG, and Bk-VPG graphs, for each value k. [ABSTRACT FROM AUTHOR]

Published
2023
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6. The Complexity of Helly-B1-EPG graph Recognition.

Author

Bornstein, Claudson F., Golumbic, Martin Charles, Santos, Tanilson D., Souza, Uéverton S., and Szwarcfiter, Jayme L.

Subjects
*GRAPH theory, *GEOMETRIC analysis, *ALGORITHMS, *PROBABILISTIC databases, *QUANTUM theory
Abstract

Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the intersection graph class of edge paths on a grid. An EPG graph G is a graph that admits a representation where its vertices correspond to paths in a grid Q, such that two vertices of G are adjacent if and only if their corresponding paths in Q have a common edge. If the paths in the representation have at most k bends, we say that it is a Bk-EPG representation. A collection C of sets satisfies the Helly property when every sub-collection of C that is pairwise intersecting has at least one common element. In this paper, we show that given a graph G and an integer k, the problem of determining whether G admits a Bk-EPG representation whose edge-intersections of paths satisfy the Helly property, so-called Helly-Bk-EPG representation, is in NP, for every k bounded by a polynomial function of jV (G)j. Moreover, we show that the problem of recognizing Helly-B1-EPG graphs is NP-complete, and it remains NP-complete even when restricted to 2-apex and 3-degenerate graphs. [ABSTRACT FROM AUTHOR]

Published
2020

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