An O(n) Time Algorithm to Find Cubic Subgraph in a Halin Graph

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.380-384.1318 ◽

2013 ◽

Vol 380-384 ◽

pp. 1318-1322

Author(s):

Ding Jun Lou ◽

Jun Fu Liu

Keyword(s):

Time Algorithm ◽

Halin Graph ◽

Complete Problem ◽

Np Complete ◽

Regular Subgraph ◽

General Graphs

The 3-Regular Subgraph Problem is: Given a graph G = (V, E), can we find a subgraph H = (V, E) in G such that for each vertex u in V, , where is the degree of u in H? This problem is an NP-complete problem for general graphs. In this paper, we design an O(n) time algorithm to solve The 3-Regular Subgraph Problem for a Halin graph H, where n is the number of vertices of H. Given a Halin graph H, if there is a cubic subgraph G in H, then our algorithm will find G and give an answer Yes, otherwise our algorithm will give an answer No. We also prove the correctness of this algorithm.

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Polynomial time solution to NP-complete problem Hamiltonian cycle (P=NP Proof)

10.31219/osf.io/2gskv ◽

2021 ◽

Author(s):

Yasaman KalantarMotamedi

Keyword(s):

Computer Science ◽

Polynomial Time ◽

Hamiltonian Cycle ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Rigorous Proof ◽

Complete Problem ◽

Time Solution ◽

Np Complete

P vs NP is one of the open and most important mathematics/computer science questions that has not been answered since it was raised in 1971 despite its importance and a quest for a solution since 2000. P vs NP is a class of problems that no polynomial time algorithm exists for any. If any of the problems in the class gets solved in polynomial time, all can be solved as the problems are translatable to each other. One of the famous problems of this kind is Hamiltonian cycle. Here we propose a polynomial time algorithm with rigorous proof that it always finds a solution if there exists one. It is expected that this solution would address all problems in the class and have a major impact in diverse fields including computer science, engineering, biology, and cryptography.

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Uniquely monopolar-partitionable block graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2073 ◽

2014 ◽

Vol Vol. 16 no. 2 (PRIMA 2013) ◽

Author(s):

Xuegang Chen ◽

Jing Huang

Keyword(s):

Polynomial Time Algorithm ◽

Time Algorithm ◽

Special Issue ◽

Block Graphs ◽

International Audience ◽

Split Graphs ◽

Np Complete ◽

Vertex Partitions ◽

General Graphs

Special issue PRIMA 2013 International audience As a common generalization of bipartite and split graphs, monopolar graphs are defined in terms of the existence of certain vertex partitions. It has been shown that to determine whether a graph has such a partition is NP-complete for general graphs and polynomial for several classes of graphs. In this paper, we investigate graphs that admit a unique such partition and call them uniquely monopolar-partitionable graphs. By employing a tree trimming technique, we obtain a characterization of uniquely monopolar-partitionable block graphs. Our characterization implies a polynomial time algorithm for recognizing them.

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A linear time algorithm for minimum equitable dominating set in trees

Proyecciones (Antofagasta) ◽

10.22199/issn.0717-6279-4552 ◽

2021 ◽

Vol 40 (4) ◽

pp. 805-814

Author(s):

Sohel Rana ◽

Sk. Md. Abu Nayeem

Keyword(s):

Linear Time ◽

Dominating Set ◽

Domination Number ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Minimum Cardinality ◽

Np Complete ◽

General Graphs

Let G = (V, E) be a graph. A subset De of V is said to be an equitable dominating set if for every v ∈ V \ De there exists u ∈ De such that uv ∈ E and |deg(u) − deg(v)| ≤ 1, where, deg(u) and deg(v) denote the degree of the vertices u and v respectively. An equitable dominating set with minimum cardinality is called the minimum equitable dominating set and its cardinality is called the equitable domination number and it is denoted by γe. The problem of finding minimum equitable dominating set in general graphs is NP-complete. In this paper, we give a linear time algorithm to determine minimum equitable dominating set of a tree.

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An Efficient Algorithm to Solve the Conditional Covering Problem on Trapezoid Graphs

ISRN Discrete Mathematics ◽

10.5402/2011/213084 ◽

2011 ◽

Vol 2011 ◽

pp. 1-10

Author(s):

Akul Rana ◽

Anita Pal ◽

Madhumangal Pal

Keyword(s):

Time Algorithm ◽

Chordal Graphs ◽

Covering Problem ◽

Shortest Distance ◽

Trapezoid Graph ◽

Np Complete ◽

Conditional Covering ◽

General Graphs ◽

Special Case ◽

Trapezoid Graphs

Let G=(V,E) be a simple connected undirected graph. Each vertex v∈V has a cost c(v) and provides a positive coverage radius R(v). A distance duv is associated with each edge {u,v}∈E, and d(u,v) is the shortest distance between every pair of vertices u,v∈V. A vertex v can cover all vertices that lie within the distance R(v), except the vertex itself. The conditional covering problem is to minimize the sum of the costs required to cover all the vertices in G. This problem is NP-complete for general graphs, even it remains NP-complete for chordal graphs. In this paper, an O(n2) time algorithm to solve a special case of the problem in a trapezoid graph is proposed, where n is the number of vertices of the graph. In this special case, duv=1 for every edge {u,v}∈E, c(v)=c for every v∈V(G), and R(v)=R, an integer >1, for every v∈V(G). A new data structure on trapezoid graphs is used to solve the problem.

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Interdicting facilities in tree networks

Top ◽

10.1007/s11750-021-00600-6 ◽

2021 ◽

Author(s):

Nicolas Fröhlich ◽

Stefan Ruzika

Keyword(s):

Objective Function ◽

Polynomial Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Solution Algorithm ◽

Graph Class ◽

Np Complete ◽

Number Of Customers ◽

General Graphs ◽

Theoretical Results

AbstractThis article investigates a network interdiction problem on a tree network: given a subset of nodes chosen as facilities, an interdictor may dissect the network by removing a size-constrained set of edges, striving to worsen the established facilities best possible. Here, we consider a reachability objective function, which is closely related to the covering objective function: the interdictor aims to minimize the number of customers that are still connected to any facility after interdiction. For the covering objective on general graphs, this problem is known to be NP-complete (Fröhlich and Ruzika In: On the hardness of covering-interdiction problems. Theor. Comput. Sci., 2021). In contrast to this, we propose a polynomial-time solution algorithm to solve the problem on trees. The algorithm is based on dynamic programming and reveals the relation of this location-interdiction problem to knapsack-type problems. However, the input data for the dynamic program must be elaborately generated and relies on the theoretical results presented in this article. As a result, trees are the first known graph class that admits a polynomial-time algorithm for edge interdiction problems in the context of facility location planning.

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Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽

10.3390/math9030293 ◽

2021 ◽

Vol 9 (3) ◽

pp. 293

Author(s):

Xinyue Liu ◽

Huiqin Jiang ◽

Pu Wu ◽

Zehui Shao

Keyword(s):

Linear Time ◽

Minimum Weight ◽

Domination Number ◽

Time Algorithm ◽

Simple Graph ◽

Linear Time Algorithm ◽

Domination Problem ◽

Np Complete ◽

Chordal Bipartite Graphs ◽

Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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k-Efficient domination: Algorithmic perspective

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830922500513 ◽

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.

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