29802636
Two classes of edge domination in graphs
Let γ's(G) (γ'l(G), resp.) be the number of (local) signed edge domination of a graph G [B. Xu, On signed edge domination numbers of graphs, Discrete Math. 239 (2001) 179-189]. In this paper, we prove mainly that γ's(G) ≤ ⌊11/6n - 1⌋ and γ's(G) ≤ 2n - 4 hold for any graph G of order n (n ≥ 4), and pose several open problems and conjectures.
24660543
A note on the lower bounds of signed domination number of a graph
Abstract Let G = (V, E) be a graph. For a function f : V → {−1, 1}, the weight of f is w(f) = Σv ∈ V f(v). For a vertex v in V, we define f[v] = Σu∈N[v] f(u). A signed dominating function of G is a function f : V → {−1, 1} such that f[v] ⩾ 1 for all v ∈ V. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. A signed dominating function of a weight γs(G) we call a γs-function of G. In this paper, we study the signed domination problem of general graph, and obtain some lower bounds of the signed domination number of a graph, and show that these lower bounds are sharp, and extend a result in Dunbar et al. (1995).
30641096
On the fixed edge of planar graphs with minimum degree five
Abstract An edge e of a finite and simple graph G is called a fixed edge of G if G - e + e ′ ≅ G implies e ′ = e . In this paper, we show that planar graphs with minimum degree 5 contain fixed edges, from which we prove that a class of planar graphs with minimum degree one is edge reconstructible.
16379968
On Structure of Some Plane Graphs with Application to Choosability
A graph G=(V, E) is (x, y)-choosable for integers xy?1 if for any given family {A(v)?v?V} of sets A(v) of cardinality x, there exists a collection {B(v)?v?V} of subsets B(v)?A(v) of cardinality y such that B(u)?B(v)=? whenever uv?E(G). In this paper, structures of some plane graphs, including plane graphs with minimum degree 4, are studied. Using these results, we may show that if G is free of k-cycles for some k?{3, 4, 5, 6}, or if any two triangles in G have distance at least 2, then G is (4m, m)-choosable for all nonnegative integers m. When m=1, (4m, m)-choosable is simply 4-choosable. So these conditions are also sufficient for a plane graph to be 4-choosable.
28979330
Connected Factors and Spanning Trees in Graphs
Abstract. Let G be a graph, and g, f, f′ be positive integer-valued functions defined on V(G). If an f′-factor of G is a spanning tree, we say that it is f′-tree. In this paper, it is shown that G contains a connected (g, f+f′−1)-factor if G has a (g, f)-factor and an f′-tree.
37612124
Minimally circular-imperfect graphs with a major vertex
In this note, we characterize the minimally circular-imperfect graphs that have a major vertex.
31739418
Geometric Algorithms for Density-based Data Clustering
Data clustering is a fundamental problem arising in many practical applications. In this paper, we present new geometric approximation and exact algorithms for the density-based data clustering problem in d-dimensional space ℝd (for any constant integer d ≥ 2). Previously known algorithms for this problem are efficient only when the specified range around each input point, called the δ-neighborhood, contains on average a constant number of input points. Different distributions of the input data points have significant impact on the efficiency of these algorithms. In the worst case when the data points are highly clustered, these algorithms run in quadratic time, although such situations might not occur very frequently on real data. By using computational geometry techniques, we develop faster approximation and exact algorithms for the density-based data clustering problem in ℝd. In particular, our approximation algorithm based on the ∊-fuzzy distance function takes O(n log n) time for any given fixed value ∊>0, and our exact algorithms take sub-quadratic time. The running times and output quality of our algorithms do not depend on any particular data distribution. We believe that our fast approximation algorithm is of considerable practical importance, while our sub-quadratic exact algorithms are more of theoretical interest. We implemented our approximation algorithm and the experimental results show that our approximation algorithm is efficient on arbitrary input point sets.