| Project/Area Number |
16540115
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | BUNKYO University |
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Principal Investigator |
ERA Hiroshi BUNKYO University, Information and communication, Professor, 情報学部, 教授 (60185147)
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| Co-Investigator(Kenkyū-buntansha) |
NEMOTO Toshio BUNKYO University, Information and Communication, Assistant Professor, 情報学部, 助教授 (40286026)
HOTTA Keisuke BUNKYO University, Information and Communication, Lecturer, 情報学部, 講師 (80327022)
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| Project Period (FY) |
2004 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 2005: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2004: ¥2,100,000 (Direct Cost: ¥2,100,000)
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| Keywords | Applied Mathematics / Graph Theory / Partially ordered set / upper bound graph / poset / 離散数学 / 組合せ論 |
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| Research Abstract |
In this research, we consider the characterizations of upper bound graphs of posets from various aspects. For a poset P, the upper bound graph of P is the graph G_p=(X,E), where an edge uv belongs to E iff there exists an element m of X such that m is an upper bound of u and v. F.R.McMorris and T.Zaslavsky show a characterization of upper bound graphs using the clique covering conception. Here we give different characterizations by constructive method and also by forbidden substructures
An upper bound graph can be transformed into a nova by contractions and a nova can be transformed into an upper bound graph by splits. Where nova is a graph class obtained from a star K_<1.n> by replacing each edge with a complete graph with at least two edges. By these results, we get a characterization on upper bound graphs as follows.
Let G be a connected graph. G is an upper bound graph if the graph obtained by successive contractions of adjacent non-simplicial vertices u and v satisfying the following conditions is a nova :
(1)u and v are adjacent to a simplicial vertex of G.
(2)there exist no pair of non-simplicial adjacent vertices x andy which are not adjacent to simlicial vertices of G satisfying ceirtain conditions.
The second result is on the characterization of some restricted class of upper bound graphs by concept of forbidden subset in posets. For a poset P, a poset Q is an m-subposet of P iff (1)Q is a subposet of P and, (2)for any pair x,y of Q, x,y <=m for some m in P then there exists m' in Q with x,y <=m'. This definition is introduced by D.D.Scott in 1986. Using this concept we give the following results,
(1)G is a split upper bound graph iff the canonical poset of G contain no poset P_<2K2> as an m-subposet.
(2)G is a threshold upper bound graph iff the canonical poset of G contain no 2K_2 or P_w as m-subposets.
(3)G is difference upper bound graph iff the canonical poset of G contain P_<2K2> or P^as m-subposets.
where P_w and P^are certain elementary classes of posets.
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