Domination number and minimum dominating sets in pseudofractal scale-free web and Sierpiński graph

Abstract

The minimum dominating set (MDS) problem is a fundamental subject of theoretical computer science, and has found vast applications in different areas, including sensor networks, protein interaction networks, and structural controllability. However, the determination of the size of a MDS and the number of all MDSs in a general network is NP-hard, and it thus makes sense to seek particular graphs for which the MDS problem can be solved analytically. In this paper, we study the MDS problem in the pseudofractal scale-free web and the Sierpiński graph, which have the same number of vertices and edges. For both networks, we determine explicitly the domination number, as well as the number of distinct MDSs. We show that the pseudofractal scale-free web has a unique MDS, and its domination number is only half of that for the Sierpiński graph, which has many MDSs. We argue that the scale-free topology is responsible for the difference of the size and number of MDSs between the two studied graphs, which in turn indicates that power-law degree distribution plays an important role in the MDS problem and its applications in scale-free networks.

Spanning Trees of Recursive Scale-Free Graphs

“…It is a small-world complex network with hierarchical structure, a scale-free degree distribution with power law exponent γ ≈ 2.57, and a high clustering coefficient of C = 4/5, matching several characteristic benchmarks of real-life complex nets. Its recursive nature and the finite articulation have enabled exact analytical studies of the statistics of cycles [5], diffusion [6], percolation [3], spectral properties [7], and minimum dominating sets [8], shedding light on the analogous properties in real-life complex nets.…”

Variants of the Domination Number for Flower Snarks

“…Other than trivial results such as for paths or cycles, perhaps the most famous result is the sprawling effort over a 27 year period [18,14,8,34,2,12] to provide a complete characterisation of domination numbers for grid graphs G(n, m) of all possible sizes, consisting of 23 special cases before settling into a standard formula for n, m ≥ 16. Other results for domination include generalized Petersen graphs [41,23,39], Cartesian products involving cycles [26,9,1], King graphs [40], Latin square graphs [27], hypercubes [3], Sierpiński graphs [32], Knödel graphs [38], and various graphs from chemistry [25,30], among others.…”

Some Combinatorial Problems in Power-Law Graphs

“…In the past decade, these problems have become very active and have been popular research objects. Many authors have devoted their efforts to developing algorithms for the problems associated with maximum matchings [15][16][17][18][19][20], MISs [21][22][23], as well as MDSs [24][25][26][27][28]. Although scientists have made a concerted effort, solving these problems is an important challenge and often computationally difficult.…”

“…This nontrivial scale-free structure is a fundamental concept in the study of the emerging network sciences. Many previous studies have shown that the scale-free topology plays an important role in various structural [36], combinatorial [13,19,[26][27][28], and dynamical [37][38][39] properties of a graph. In the context of combinatorial aspect, it has been shown that compared with non-scale-free graphs, in scalefree networks, both the matching number [13] and the number of maximum matchings [19] are significantly smaller.…”