On Distances of Posets with the Same Upper Bound Graphs

In this paper, we consider transformations for posets with the same upper bound graph. Those $poset_{8}$ can be $tran8formed$ into each other by a finite sequence of two kinds of transformations, called addition and deletion of an order relation. This result induces a characterization on unique upper bound graphs. We deal with the distance of those posets that have the same upper bound graph. Introduction In this paper, we consider finite undirected simple graphs. Let $P=(X, \leq)$ be a poset. The upper bound graph (UB-graph) of $P$ is the graph $UB(P)$ over $X$ obatined by joining a pair of distinct elements $u$ and $v$ in $X$ whenever there exists $m\in X$ such that $u,$ $v\leq m$ . We say that a graph $G$ is a UB-graph if there exists a poset whose upper bound graph is isomorphic to $G$ . These concepts were introduced by F.R. McMorris and T. Zaslavsky [2]. The total ordered set with $n$ elements is not isomorphic to the height-one poset with a unique maximal element and $n-1$ minimal elements if $n\geq 3$ , but their UB-graphs are isomorphic. Thus a natural question arises; how are those two posets that have the same UB-graphs related. In this paper, we shall answer this quetion, introducing two kinds of transformations of such posets. 1. Upper bound graphs In this section, we introduce two kinds of transformations for those posets that have the same UB-graph. A characterization of upper bound graphs can be found in [2] as follows: A clique in a graph $G$ is a maximal set of vertices which induces a complete subgraph in $G$ . A family $C$ of complete subgraphs is said to edge-cover $G$ if for each edge $uv\in E(G)$ , there exists $C\in C$ such that $u,$ $v\in C$ . 1991 Mathematics Subject Classffication: $05C12,05C75$ , 06A06