Abstract An ordered circular permutation S of u's and v' s is called an ordered circular sequence of u' s and v' s. A kernel of a digraph G=(V,A) is an independent subset of V, say K, such that for any...Abstract An ordered circular permutation S of u's and v' s is called an ordered circular sequence of u' s and v' s. A kernel of a digraph G=(V,A) is an independent subset of V, say K, such that for any vertex vi in V\K there is an arc from vi to a vertex vj in K. G is said to be kernel-perfect (KP) if every induced subgraph of G has a kernel. G is said to be kernel-perfect-critical (KPC) if G has no kernel but every proper induced subgraph of G has a kernel. The digraph G=(V,A)= $\overrightarrow {C_n }$ (j1,j2,...,jk) is defined by: V(G)={0,1,...,nm1}, A(G)={uv | vmuLj, (mod n) for 1 hihk}.In an earlier work, we investigated the digraph G= $\overrightarrow {C_n }$ (1-'d,-2d,-3d,...,-sd), denoted by G(n,d,r,s), where '=1 for d>1 or '=0 for d=1, and n,d,r,s are positive integers with (n,d)=r and n=mr, and gave some necessary and sufficient conditions for G(n,d,r,s) with rS3 and s=1 to be KP or KPC.In this paper, we prove a combinatorial theorem on ordered circular sequences of n1 u's and n2 v's. By using the theorem, we prove that, if (n,d)=rS2 and sS2, then G(n,d,r,s) is a KP graph.展开更多
This paper presents a new effcient algorithm for exactly computing the halfspace depth contours based on the idea of a circular sequence. Unlike the existing methods, the proposed algorithm segments the unit sphere di...This paper presents a new effcient algorithm for exactly computing the halfspace depth contours based on the idea of a circular sequence. Unlike the existing methods, the proposed algorithm segments the unit sphere directly relying on the permutations that correspond to the projections of observations onto some unit directions, without having to use the technique of parametric programming.Some data examples are also provided to illustrate the performance of the proposed algorithm.展开更多
基金Supported by the National Natural Sciences Foundation of China (No.19831080).
文摘Abstract An ordered circular permutation S of u's and v' s is called an ordered circular sequence of u' s and v' s. A kernel of a digraph G=(V,A) is an independent subset of V, say K, such that for any vertex vi in V\K there is an arc from vi to a vertex vj in K. G is said to be kernel-perfect (KP) if every induced subgraph of G has a kernel. G is said to be kernel-perfect-critical (KPC) if G has no kernel but every proper induced subgraph of G has a kernel. The digraph G=(V,A)= $\overrightarrow {C_n }$ (j1,j2,...,jk) is defined by: V(G)={0,1,...,nm1}, A(G)={uv | vmuLj, (mod n) for 1 hihk}.In an earlier work, we investigated the digraph G= $\overrightarrow {C_n }$ (1-'d,-2d,-3d,...,-sd), denoted by G(n,d,r,s), where '=1 for d>1 or '=0 for d=1, and n,d,r,s are positive integers with (n,d)=r and n=mr, and gave some necessary and sufficient conditions for G(n,d,r,s) with rS3 and s=1 to be KP or KPC.In this paper, we prove a combinatorial theorem on ordered circular sequences of n1 u's and n2 v's. By using the theorem, we prove that, if (n,d)=rS2 and sS2, then G(n,d,r,s) is a KP graph.
基金supported by the National Natural Science Foundation of China under Grant No.11461029the Natural Science Foundation of Jiangxi Province under Grant Nos.20142BAB211014+5 种基金20132BAB21101520122BAB20102320133BCB23014the Youth Science Fund Project of Jiangxi provincial education department under Grant Nos.GJJ14350GJJ14449KJLD13033
文摘This paper presents a new effcient algorithm for exactly computing the halfspace depth contours based on the idea of a circular sequence. Unlike the existing methods, the proposed algorithm segments the unit sphere directly relying on the permutations that correspond to the projections of observations onto some unit directions, without having to use the technique of parametric programming.Some data examples are also provided to illustrate the performance of the proposed algorithm.
