In a graph G = (V, E), a non-empty set A of k distinct vertices, is called a k-attack on G. The vertices in the set A are considered to be under attack. A set D subset of V can defend or counter the attack A on G if t...
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In a graph G = (V, E), a non-empty set A of k distinct vertices, is called a k-attack on G. The vertices in the set A are considered to be under attack. A set D subset of V can defend or counter the attack A on G if there exists a one-to-one function f : A bar right arrow D, such that either f (u) = u or there is an edge between u, and its image f (u), in G. A set D is called a k-defensive dominating set if it defends against any k-attack on G. Given a graph G = (V, E), the minimum k-defensive domination problem requires us to compute a minimum cardinality k-defensive dominating set of G. When k is not fixed, it is co-NP-hard to decide if D subset of V is a k-defensive dominating set. However, when k is fixed, the decision version of the problem is NP-complete for general graphs. On the positive side, the problem can be solved in linear time when restricted to paths, cycles, co-chain, and threshold graphs for any k. This paper mainly focuses on the problem when k > 0 is fixed. We prove that the decision version of the problem remains NPcomplete for bipartite graphs;this answers a question asked by Ekim et al. (Discrete Math. 343 (2) (2020)). We establish a lower and upper bound on the approximation ratio for the problem. Further, we show that the minimum k-defensive domination problem is APX-complete for bounded degree graphs. On the positive side, we show that the problem is efficiently solvable for complete bipartite graphs for any k > 0. Towards the end, we study a relationship between the defensive domination number and another well-studied domination parameter. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
A pangenome graph can represent the genomes of multiple individuals simultaneously. It provides a more comprehensive reference and overcomes the allele bias caused by linear reference genome, which is becoming a power...
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Triangle counting is a critical problem in graph theory and network analysis, with widespread applications in social network analysis, network science, and bioinformatics. Dynamic graphs are extensively utilized in va...
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Stochastic optimization algorithms are widely used for large-scale data analysis due to their low per-iteration costs, but they often suffer from slow asymptotic convergence caused by inherent variance. Variance-reduc...
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A graph G is k-vertex-critical if chi(G) = k but chi(G-v)= 1. center dot There are only finitely many k-vertex-critical (co-gem, P5, P3+cP2)-free graphs for all k >= 1 and c >= 0. To prove the latter result, we ...
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A graph G is k-vertex-critical if chi(G) = k but chi(G-v)graph isomorphic to H-1 or H-2. We show that there are only finitely many k-vertex-critical (co-gem, H)-free graphs for all k when H is any graph of order 4 by showing finiteness in the three remaining open cases, those are the cases when H is 2P(2), K-3+P-1, and K-4. For the first two cases we actually prove the stronger results: center dot There are only finitely many k-vertex-critical (co-gem, paw+P1)-free graphs for all k >= 1. center dot There are only finitely many k-vertex-critical (co-gem, P5, P3+cP2)-free graphs for all k >= 1 and c >= 0. To prove the latter result, we employ a novel application of Sperner's Theorem on the number of antichains in a partially ordered set. Our result for K-4 uses exhaustive computer search and is proved by showing the stronger result that every (co-gem, K-4)-free graph is 4-colourable. Our results imply the existence of simple polynomial-time certifying algorithms to decide the k-colourability of (co-gem, H)-free graphs for all k and all H of order 4 by searching the vertex-critical graphs as induced subgraphs.
The graph coloring problem, a fundamental NP-hard challenge, has numerous applications in scheduling, register allocation, and network optimization. Traditional sequential algorithms for graph coloring are computation...
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We give an algorithm to morph planar graph drawings that achieves small grid size at the expense of allowing a constant number of bends on each edge. The input is an n-vertex planar graph and two planar straight-line ...
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This study aimed to leverage graph information, particularly Rhetorical Structure Theory (RST) and Co-reference (Coref) graphs, to enhance the performance of our baseline summarization models. Specifically, we experim...
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We introduce a new graph sparsification method that targets the neighborhood information available for each node. Our approach is motivated by the fact that neighborhood information is used by several mining and learn...
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graph coarsening is an important step for many multi-level algorithms, most notably graph partitioning. However, such methods often utilize an iterative approach, where a new coarser graph representation is explicitly...
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