Algorithmic aspect on the minimum (weighted) doubly resolving set problem of graphs

Changhong Lu (), Qingjie Ye () and Chengru Zhu
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Changhong Lu: East China Normal University
Qingjie Ye: East China Normal University
Chengru Zhu: East China Normal University

Journal of Combinatorial Optimization, 2022, vol. 44, issue 3, No 34, 2029-2039

Abstract: Abstract Let G be a simple graph, where each vertex has a nonnegative weight. A vertex subset S of G is a doubly resolving set (DRS) of G if for every pair of vertices u, v in G, there exist $$x,y\in S$$ x , y ∈ S such that $$d(x,u)-d(x,v)\ne d(y,u)-d(y,v)$$ d ( x , u ) - d ( x , v ) ≠ d ( y , u ) - d ( y , v ) . The minimum weighted doubly resolving set (MWDRS) problem is finding a doubly resolving set with minimum total weight. We establish a linear time algorithm for the MWDRS problem of all graphs in which each block is complete graph or cycle. Hence, the MWDRS problems for block graphs and cactus graphs can be solved in linear time. We also prove that k-edge-augmented tree (a tree with additional k edges) with minimum degree $$\delta (G)\ge 2$$ δ ( G ) ≥ 2 admits a doubly resolving set of size at most $$2k+1$$ 2 k + 1 . This implies that the DRS problem on k-edge-augmented tree can be solved in $$O(n^{2k+3})$$ O ( n 2 k + 3 ) time.

Keywords: Doubly resolving set; Block graph; Cactus graph; k-edge-augmented trees (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10878-020-00575-3

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