We prove that for a finite planar space \( S = (\mathcal{P}, \mathcal{L}, \mathcal{H}) \) with no disjoint planes and with a constant number of planes on a line, the number \( \ell \) of lines is greater than or equal to the number \( c \) of planes, and the equality holds true if and only if \( S \) is either the finite Desarguesian 4-dimensional projective space \( PG(4,q) \), or the complete graph \( K_5 \).

A \((p, q)\)-graph \( G \) is said to be \textbf{edge graceful} if the edges can be labeled by \( 1, 2, \ldots, q \) so that the vertex sums are distinct, mod \( p \). It is shown that if a tree \( T \) is edge-graceful, then its order must be odd. Lee conjectured that all trees of odd orders are edge-graceful. J. Mitchem and A. Simoson introduced the concept of super edge-graceful graphs, which is a stronger concept than edge-graceful for some classes of graphs.

A graph \( G = (V, E) \) of order \( p \) and size \( q \) is said to be \textbf{super edge-graceful} (SEG) if there exists a bijection
\[
\text{f: E} \to
\begin{cases}
\{0, +1, -1, +2, -2, \ldots, \frac{q-1}{2}, -\frac{q-1}{2}\} & \text{if } q \text{ is odd} \\
\{+1, -1, +2, -2, \ldots, \frac{q}{2}, -\frac{q}{2}\} & \text{if } q \text{ is even}
\end{cases}
\]
such that the induced vertex labeling \( f^* \) defined by \( f^*(u) = \sum \{ f(u, v) : (u, v) \in E \} \) has the property:
\[
f^*: V \to
\begin{cases}
\{0, +1, -1, \ldots, +\frac{p-1}{2}, -\frac{p-1}{2}\} & \text{if } p \text{ is odd} \\
\{+1, -1, \ldots, +\frac{p}{2}, -\frac{p}{2}\} & \text{if } p \text{ is even}
\end{cases}
\]
is a bijection.

The conjecture is still unsettled. In this paper, we first characterize spiders of even orders which are not SEG. We then exhibit some spiders of even orders which are SEG of diameter at most four. By the concepts of the irreducible part of an even tree \( T \), we show that an infinite number of spiders of even orders are SEG. Finally, we provide some conjectures for further research.

In this paper, we shall consider acquisition sequences of a graph. The formation of each acquisition sequence is a process that creates an independent set. Each acquisition sequence is a sequence of “acquisitions” which are defined on a graph \( G \) for which each vertex originally has a value of one associated with it. In an acquisition, a vertex transfers all of its value to an adjacent vertex with equal or greater value. For an acquisition sequence, one continues until no more acquisitions are possible. The parameter \( a(G) \) is defined to be the minimum possible number of vertices with a nonzero value at the conclusion of such an acquisition sequence. Clearly, if \( S \) is a set of vertices with nonzero values at the end of some acquisition sequence, then \( S \) is independent, and we call such a set \( S \) an acquisition set. We show that for a given graph \( G \), “Is \( a(G) = 1 \)” is NP-complete, and describe a linear time algorithm to determine the acquisition number of a caterpillar.