Complete-Subgraph-Transversal-Sets problem on bounded treewidth graphs

2021 ◽  
Author(s):  
Ke Liu ◽  
Mei Lu
Keyword(s):  
Complete Subgraph ◽  
Bounded Treewidth

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

scholarly journals Discovering the maximum k-clique on social networks using bat optimization algorithm

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Akram Khodadadi ◽  
Shahram Saeidi
Keyword(s):  
Genetic Algorithm ◽  
Social Networks ◽  
Social Network ◽  
Coding Theory ◽  
Evaluation Criteria ◽  
Convergence Speed ◽  
Optimization Approach ◽  
Computational Results ◽  
Complete Subgraph ◽  
Np Complete

AbstractThe k-clique problem is identifying the largest complete subgraph of size k on a network, and it has many applications in Social Network Analysis (SNA), coding theory, geometry, etc. Due to the NP-Complete nature of the problem, the meta-heuristic approaches have raised the interest of the researchers and some algorithms are developed. In this paper, a new algorithm based on the Bat optimization approach is developed for finding the maximum k-clique on a social network to increase the convergence speed and evaluation criteria such as Precision, Recall, and F1-score. The proposed algorithm is simulated in Matlab® software over Dolphin social network and DIMACS dataset for k = 3, 4, 5. The computational results show that the convergence speed on the former dataset is increased in comparison with the Genetic Algorithm (GA) and Ant Colony Optimization (ACO) approaches. Besides, the evaluation criteria are also modified on the latter dataset and the F1-score is obtained as 100% for k = 5.

Hardness and tractability of the γ-Complete Subgraph problem

2021 ◽  
Vol 169 ◽  
pp. 106105
Author(s):  
Ambroise Baril ◽  
Riccardo Dondi ◽  
Mohammad Mehdi Hosseinzadeh
Keyword(s):  
Complete Subgraph

scholarly journals C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

Algorithmica ◽  
2021 ◽  
Author(s):  
Giordano Da Lozzo ◽  
David Eppstein ◽  
Michael T. Goodrich ◽  
Siddharth Gupta
Keyword(s):  
Dual Graph ◽  
Graph Visualization ◽  
Closed Disk ◽  
Running Time ◽  
Planar Embedding ◽  
Bounded Treewidth ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
The One ◽  
Clustered Planarity

AbstractFor a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time $$O(r(n^2) + n^2)$$ O ( r ( n 2 ) + n 2 ) for general, possibly non-flat, instances.

Orthogonal planarity testing of bounded treewidth graphs

2022 ◽  
Vol 125 ◽  
pp. 129-148
Author(s):  
Emilio Di Giacomo ◽  
Giuseppe Liotta ◽  
Fabrizio Montecchiani
Keyword(s):  
Bounded Treewidth

scholarly journals A Canonical Ramsey Theorem for Exactly m-Coloured Complete Subgraphs

2013 ◽  
Vol 23 (1) ◽  
pp. 102-115 ◽  
Author(s):  
TEERADEJ KITTIPASSORN ◽  
BHARGAV P. NARAYANAN
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Infinite Subset ◽  
Complete Subgraph ◽  
Ramsey Theorem ◽  
Distinguished Vertex ◽  
Edge Colourings ◽  
Distinct Colour

Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an exactly m-coloured complete subgraph for every natural number m or there exists an infinite subset X ⊂ $\mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex v in X such that X\{v} is 1-coloured and each edge between v and X\{v} has a distinct colour (all different to the colour used on X\{v}). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding exactly m-coloured complete subgraphs in colourings with finitely many colours.

Neighbor sum distinguishing total coloring of graphs with bounded treewidth

2018 ◽  
Vol 36 (1) ◽  
pp. 23-34
Author(s):  
Miaomiao Han ◽  
You Lu ◽  
Rong Luo ◽  
Zhengke Miao
Keyword(s):  
Total Coloring ◽  
Bounded Treewidth

scholarly journals The Complexity of Computing Maximin Share Allocations on Graphs

2020 ◽  
Vol 34 (02) ◽  
pp. 2006-2013
Author(s):  
Gianluigi Greco ◽  
Francesco Scarcello
Keyword(s):  
A Priori ◽  
Utility Functions ◽  
Indivisible Goods ◽  
Bounded Treewidth ◽  
Underlying Graph ◽  
Low Degree

Maximin share is a compelling notion of fairness proposed by Buddish as a relaxation of more traditional concepts for fair allocations of indivisible goods. In this paper we consider this notion within a setting where bundles of goods must induce connected subsets over an underlying graph. This setting received much attention in earlier literature, and our study answers a number of questions that were left open. First, we show that computing maximin share allocations is FΔ2P-complete, even when focusing on consistent scenarios, that is, where such allocations are a-priori guaranteed to exist. Moreover, the problem remains intractable if all agents have the same type, i.e., have the same utility functions, and if either the values returned by the utility functions are polynomially bounded, or the underlying graphs have a low degree of cyclicity (more precisely, have bounded treewidth). However, if these conditions hold all together, then computing maximin share allocations (or checking that none exists) becomes tractable. The result is established via machineries based on logspace alternating machines that use partial representations of connected bundles, which are interesting in their own.

scholarly journals (Star, k)-colourings of graphs with bounded treewidth

2020 ◽  
Author(s):  
C. A. Weffort-Santos ◽  
L. L. C. Pedrosa
Keyword(s):  
Open Problem ◽  
Time Algorithm ◽  
Graph Colouring ◽  
Bounded Treewidth ◽  
Colour Class

We study a generalization of graph colouring define as follows. Given a graph G, a (star, k)-colouring of G is a colouring c : V(G) → {1, ..., k} such that every colour class induces a star. We propose an O*(2^(O(tw))k^(tw)-time algorithm that decides whether a graph G of treewidth at most tw admits a (star, k)-colouring. This resolves an open problem posed by Angelini et al. in 2017. Our approach can be extended to other defective colouring models.

Sign in / Sign up

Export Citation Format

Share Document