
On Efficient Domination for Some Classes of HFree Bipartite Graphs
A vertex set D in a finite undirected graph G is an efficient dominatin...
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Hard Problems That Quickly Become Very Easy
A graph class is hereditary if it is closed under vertex deletion. We gi...
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Polynomial Time Efficient Construction Heuristics for Vertex Separation Minimization Problem
Vertex Separation Minimization Problem (VSMP) consists of finding a layo...
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Cluster Deletion on Interval Graphs and Split Related Graphs
In the Cluster Deletion problem the goal is to remove the minimum numbe...
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The kPower Domination Number in Some SelfSimilar Graphs
The kpower domination problem is a problem in graph theory, which has a...
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Weighted proper orientations of trees and graphs of bounded treewidth
Given a simple graph G, a weight function w:E(G)→N∖{0}, and an orientati...
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Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six
We consider the problem of counting all kvertex subgraphs in an input g...
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Lineartime Algorithms for Eliminating Claws in Graphs
Since many NPcomplete graph problems have been shown polynomialtime solvable when restricted to clawfree graphs, we study the problem of determining the distance of a given graph to a clawfree graph, considering vertex elimination as measure. CLAWFREE VERTEX DELETION (CFVD) consists of determining the minimum number of vertices to be removed from a graph such that the resulting graph is clawfree. Although CFVD is NPcomplete in general and recognizing clawfree graphs is still a challenge, where the current best algorithm for a graph G has the same running time of the best algorithm for matrix multiplication, we present lineartime algorithms for CFVD on weighted block graphs and weighted graphs with bounded treewidth. Furthermore, we show that this problem can be solved in linear time by a simpler algorithm on forests, and we determine the exact values for full kary trees. On the other hand, we show that CLAWFREE VERTEX DELETION is NPcomplete even when the input graph is a split graph. We also show that the problem is hard to approximate within any constant factor better than 2, assuming the Unique Games Conjecture.
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