7708894
Decomposing toroidal graphs into circuits and edges
Erdos et al. (Canad. J. Math. 18 (1966) 106-112) conjecture that there exists a constant d"c"e such that every simple graph on n vertices can be decomposed into at most d"c"en circuits and edges. We consider toroidal graphs, where the graphs can be embedded on the torus, and give a polynomial time algorithm to decompose the edge set of an even toroidal graph on n vertices into at most (n+3)/2 circuits. As a corollary, we get a polynomial time algorithm to decompose the edge set of a toroidal graph (not necessarily even) on n vertices into at most 3(n-1)/2 circuits and edges. This settles the conjecture for toroidal graphs.
30431957
On endo-homology of complexes of graphs
Abstract Let L be a subcomplex of a complex K . If the homomorphism from inclusion i ∗ :H q (L)→H q (K) is an isomorphism for all q ⩾ 0, then we say that L and K are endo-homologous. The clique complex of a graph G , denoted by C ( G ), is an abstract complex whose simplices are the cliques of G . The present paper is a generalization of Ivashchenko (1994) along several directions. For a graph G and a given subgraph F of G , some necessary and sufficient conditions for C ( G ) to be endo-homologous to C ( F ) are given. Similar theorems hold also for the independence complex I ( G ) of G , where I ( G ) − C ( G c ), the clique complex of the complement of G .
20988039
Two conjectures equivalent to the perfect graph conjecture
Abstract In this note, we study the graphs with the property that each of their induced subgraphs has circular clique number (defined in Section 2) the same as its clique number, prove that a graph satisfies such a property if and only if neither itself nor its complement contains an induced subgraph isomorphic to an odd cycle of length at least five, and then give two conjectures which are equivalent to the perfect graph conjecture.
12925025
The 4-Choosability of Plane Graphs without 4-Cycles'
A graphGis calledk-choosable ifkis a number such that if we give lists ofkcolors to each vertex ofGthere is a vertex coloring ofGwhere each vertex receives a color from its own list no matter what the lists are. In this paper, it is shown that each plane graph without 4-cycles is 4-choosable.
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.
37647794
On 3-colorable plane graphs without 5- and 7-cycles
In this note, it is proved that every plane graph without 5- and 7-cycles and without adjacent triangles is 3-colorable. This improves the result of [O.V. Borodin, A.N. Glebov, A. Raspaud, M.R. Salavatipour, Planar graphs without cycles of length from 4 to 7 are 3-colorable, J. Combin. Theory Ser. B 93 (2005) 303-311], and offers a partial solution for a conjecture of Borodin and Raspaud [O.V. Borodin, A. Raspaud, A sufficient condition for planar graphs to be 3-colorable, J. Combin. Theory Ser. B 88 (2003) 17-27].
18072223
Mining association rules with linguistic terms
Some problems of mining association rules with linguistic terms are discussed. First, an incremental updating algorithm of association rules with linguistic terms is presented. The collection of frequent linguistic attribute sets and its negative border along with their support count are maintained, which makes scan the entire database once at most in the process of updating association rules. The experiment shows that the updating algorithm can not only update association rules effectively but also avoid the repeated cost. Secondly, the parallel algorithm for mining association rules with linguistic terms is presented. The Boolean parallel mining algorithm is improved to discover frequent linguistic attribute sets, and the association rules with at least confidence are generated on all processors. This parallel mining algorithm has fine scale-up, size-up and speed-up.
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).
39744832
On signed edge domination numbers of graphs
Abstract Given a graph G=(V,E) , if e=uv∈E , then the closed edge-neighbourhood of e is denoted by N[e]={u′v′∈E|u′=u or v′=v} . A function f : E→{+1,−1} is called the signed edge domination function (SEDF) of G if ∑ e′∈N[e] f(e′)⩾1 for every e∈E . The signed edge domination number γ s ′(G) of G is defined as γ s ′(G)= min {∑ e∈E f(e) | f is an SEDF of G} . Let Ψ(m)= min {γ s ′(H)|H is a graph with |E(H)|=m} . In this paper, we determine the exact value of Ψ(m) for each positive integer m . That is: Ψ(m)=2 1 3 24m+25 +6m+5 6 −m, where ⌈x⌉ denotes the ceiling of x . In addition, we also characterize all connected graphs G with γ s ′(G)=|E(G)| .
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.