| Research Abstract |
We prove the following theorem : Let $m$ be a positive integer, and let $T_1, \cdots, T_q$ be $q$ disjoint rooted trees such that $|T_ i| \in \ {m, m+1\} $ and $v_i$ is the root of $T_i$ for all $ 1\leq i\leq q$. Let $P$ be a set of $|T_1|+ \cdots +|T_q|$ points in the plane in general position that contains $q$ specified points $p_1, \cdots, p_q $. Then the rooted forest $ T_1 \cup \cdots \cup T_q$ with roots $v_1, \cdots, v_q$ can be straight-line embedded onto $P$ so that each $v_i$ corresponds to $p_i$ for every $1 \le i \le q$.
In order to prove the theorem above, we prove the next theorem :
Let $m$ be a positive integer and let $S_1$, $ S_2$ and $T$ be three disjoint sets of points in the plane such that no three points of $S_1 \cup S_2 \cup T$ lie on the same lineand $|T|=(m-l)|S_1|+ m|S_2|$. Put $q=|S_1 \cup S_2|$.
Then $S_1 \cup S_2 \cup T$ can be partitioned into $q$ disjoint subsets $P_1, \cdots, P_q$ satisfying the following three conditions :
(i) ${\rm \mbox {conv}} \, (P_i) \cap {\rm \mbox {conv}} \, (P_j)= \emptyset $ for all $1 \leq i<j \leq q$ ;
(ii) S|P_i \cap (S_1 \cup S_2) |=l$ for all $1 \leq i \leq q$ ; and
(iii) $|P_i \cap T|=m-1$ if $|P_i \cap S_1|=1$, and $|P_i \cap T|=m$ if $|P_i \cap S_2|=-1$.
This partition is called a semi-balanced partition.
Our proof gives an $0 (n^4) $ time algorithm for finding the above straight-line embedding of the rooted forest $ T_1\cup \cdots \cup T_q$ of order $n=|T_1|+ \cdots +|T_q|$.
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