Linear layouts of weakly triangulated graphs

2016 ◽  
Vol 08 (03) ◽  
pp. 1650038 ◽  
Author(s):  
Asish Mukhopadhyay ◽  
S. V. Rao ◽  
Sidharth Pardeshi ◽  
Srinivas Gundlapalli
Keyword(s):  
Time Algorithm ◽  
Chordal Graphs ◽  
Sparse Graphs ◽  
Interesting Connection ◽  
Triangulated Graphs ◽  
Distance Data ◽  
Point Placement ◽  
Appl Math ◽  
Block Tree ◽  
Triangulated Graph

A graph [Formula: see text] is said to be triangulated if it has no chordless cycles of length 4 or more. Such a graph is said to be rigid if, for a valid assignment of edge lengths, it has a unique linear layout and non-rigid otherwise. Damaschke [Point placement on the line by distance data, Discrete Appl. Math. 127(1) (2003) 53–62] showed how to compute all linear layouts of a triangulated graph, for a valid assignment of lengths to the edges of [Formula: see text]. In this paper, we extend this result to weakly triangulated graphs, resolving an open problem. A weakly triangulated graph can be constructively characterized by a peripheral ordering of its edges. The main contribution of this paper is to exploit such an edge order to identify the rigid and non-rigid components of [Formula: see text]. We first show that a weakly triangulated graph without articulation points has at most [Formula: see text] different linear layouts, where [Formula: see text] is the number of quadrilaterals (4-cycles) in [Formula: see text]. When [Formula: see text] has articulation points, the number of linear layouts is at most [Formula: see text], where [Formula: see text] is the number of nodes in the block tree of [Formula: see text] and [Formula: see text] is the total number of quadrilaterals over all the blocks. Finally, we propose an algorithm for computing a peripheral edge order of [Formula: see text] by exploiting an interesting connection between this problem and the problem of identifying a two-pair in [Formula: see text]. Using an [Formula: see text] time solution for the latter problem, we propose an [Formula: see text] time algorithm for computing its peripheral edge order, where [Formula: see text] and [Formula: see text] are respectively the number of edges and vertices of [Formula: see text]. For sparse graphs, the time complexity can be improved to [Formula: see text], using the concept of handles [R. B. Hayward, J. P. Spinrad and R. Sritharan, Improved algorithms for weakly chordal graphs, ACM Trans. Algorithms 3(2) (2007) 19pp].

k-Efficient domination: Algorithmic perspective

2022 ◽  
Author(s):  
Mohsen Alambardar Meybodi
Keyword(s):  
Decision Problem ◽  
Dominating Set ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Sparse Graphs ◽  
Fpt Algorithm ◽  
Domination Problem ◽  
Efficient Dominating Set ◽  
Np Complete

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

scholarly journals On Hyper-Chordal graphs

2021 ◽  
Vol 37 (1) ◽  
pp. 119-126
Author(s):  
MIHAI TALMACIU
Keyword(s):  
Combinatorial Optimization ◽  
Linear Complexity ◽  
Optimization Algorithms ◽  
Recognition Algorithm ◽  
Chordal Graphs ◽  
Induced Subgraph ◽  
Recognition Algorithms ◽  
Triangulated Graphs ◽  
Triangulated Graph

Triangulated graphs have many interesting properties (perfection, recognition algorithms, combinatorial optimization algorithms with linear complexity). Hyper-triangulated graphs are those where each induced subgraph has a hyper-simplicial vertex. In this paper we give the characterizations of hyper-triangulated graphs using an ordering of vertices and the weak decomposition. We also offer a recognition algorithm for the hyper-triangulated graphs, the inclusions between the triangulated graphs generalizations and we show that any hyper-triangulated graph is perfect.

On the complexity of alpha conversion

2007 ◽  
Vol 72 (4) ◽  
pp. 1197-1203
Author(s):  
Rick Statman
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Vertex Coloring ◽  
Feedback Vertex Set ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
The Way

AbstractWe consider three problems concerning alpha conversion of closed terms (combinators).(1) Given a combinator M find the an alpha convert of M with a smallest number of distinct variables.(2) Given two alpha convertible combinators M and N find a shortest alpha conversion of M to N.(3) Given two alpha convertible combinators M and N find an alpha conversion of M to N which uses the smallest number of variables possible along the way.We obtain the following results.(1) There is a polynomial time algorithm for solving problem (1). It is reducible to vertex coloring of chordal graphs.(2) Problem (2) is co-NP complete (in recognition form). The general feedback vertex set problem for digraphs is reducible to problem (2).(3) At most one variable besides those occurring in both M and N is necessary. This appears to be the folklore but the proof is not familiar. A polynomial time algorithm for the alpha conversion of M to N using at most one extra variable is given.There is a tradeoff between solutions to problem (2) and problem (3) which we do not fully understand.

scholarly journals A linear time algorithm to list the minimal separators of chordal graphs

2006 ◽  
Vol 306 (3) ◽  
pp. 351-358 ◽  
Author(s):  
L. Sunil Chandran ◽  
Fabrizio Grandoni
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Linear Time Algorithm

scholarly journals Complexity issues of perfect secure domination in graphs

2021 ◽  
Vol 55 ◽  
pp. 11
Author(s):  
P. Chakradhar ◽  
P. Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Split Graphs ◽  
Secure Domination ◽  
Convex Bipartite Graphs

Let G = (V, E) be a simple, undirected and connected graph. A dominating set S is called a secure dominating set if for each u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E and (S \{v}) ∪{u} is a dominating set of G. If further the vertex v ∈ S is unique, then S is called a perfect secure dominating set (PSDS). The perfect secure domination number γps(G) is the minimum cardinality of a perfect secure dominating set of G. Given a graph G and a positive integer k, the perfect secure domination (PSDOM) problem is to check whether G has a PSDS of size at most k. In this paper, we prove that PSDOM problem is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs, planar graphs and dually chordal graphs. We propose a linear time algorithm to solve the PSDOM problem in caterpillar trees and also show that this problem is linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. Finally, we show that the domination and perfect secure domination problems are not equivalent in computational complexity aspects.

scholarly journals Graphs with equal domination and covering numbers

2019 ◽  
Vol 39 (1) ◽  
pp. 55-71 ◽  
Author(s):  
Andrzej Lingas ◽  
Mateusz Miotk ◽  
Jerzy Topp ◽  
Paweł Żyliński
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Minimum Cardinality ◽  
Covering Numbers ◽  
Intersection Points ◽  
Covering Set ◽  
Appl Math ◽  
Side Result

Abstract A dominating set of a graph G is a set $$D\subseteq V_G$$D⊆VG such that every vertex in $$V_G-D$$VG-D is adjacent to at least one vertex in D, and the domination number $$\gamma (G)$$γ(G) of G is the minimum cardinality of a dominating set of G. A set $$C\subseteq V_G$$C⊆VG is a covering set of G if every edge of G has at least one vertex in C. The covering number $$\beta (G)$$β(G) of G is the minimum cardinality of a covering set of G. The set of connected graphs G for which $$\gamma (G)=\beta (G)$$γ(G)=β(G) is denoted by $${\mathcal {C}}_{\gamma =\beta }$$Cγ=β, whereas $${\mathcal {B}}$$B denotes the set of all connected bipartite graphs in which the domination number is equal to the cardinality of the smaller partite set. In this paper, we provide alternative characterizations of graphs belonging to $${\mathcal {C}}_{\gamma =\beta }$$Cγ=β and $${\mathcal {B}}$$B. Next, we present a quadratic time algorithm for recognizing bipartite graphs belonging to $${\mathcal {B}}$$B, and, as a side result, we conclude that the algorithm of Arumugam et al. (Discrete Appl Math 161:1859–1867, 2013) allows to recognize all the graphs belonging to the set $${\mathcal {C}}_{\gamma =\beta }$$Cγ=β in quadratic time either. Finally, we consider the related problem of patrolling grids with mobile guards, and show that it can be solved in $$O(n \log n + m)$$O(nlogn+m) time, where n is the number of line segments of the input grid and m is the number of its intersection points.

scholarly journals Semipaired Domination in Some Subclasses of Chordal Graphs

2021 ◽  
Vol vol. 23 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Michael A. Henning ◽  
Arti Pandey ◽  
Vikash Tripathi
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Positive Side ◽  
Domination Problem ◽  
Np Complete ◽  
Decision Version

A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.

Sign in / Sign up

Export Citation Format

Share Document