The Parameterised Complexity of Computing the Maximum Modularity of a Graph

The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be $$\textsf {NP}$$ NP -complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to $$\textsf {FPT}$$ FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set.

TEST REGISTER INSERTION AT RTL BASED ON REDUCED BIST

Built-in self-test (BIST) method has high area overhead and long test application time. In this paper, a new BIST method is proposed at register transfer level (RTL) as a design for testability (DFT) method to modify a given RTL circuit to a reduced BIST-able RTL circuit. First, we introduce modelling method called extended R-graph to represent the register connectivity of an RTL circuit. The original register in the RTL circuit is modified into multiple input signature registers (MISRs) as test register. The selection of MISR is performed by extended minimum feedback vertex set (MFVS) algorithm that identifies a set of vertices (representing test register) which breaks all the loops of extended R-graph with minimal cost when vertices are removed. It has been proven through simulation that the proposed BIST method has lower area overhead of 32.9% on average and achieves comparably high fault coverage compared to the previous method, concurrent BIST using ITC'99 benchmark circuits.

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.