The graph resulting from contracting edge \( e \) is denoted \( G/e \). An edge \( e \) is radius-essential if \( rad(G/e) < rad(G) \). Let \( c_r(G) \) denote the number of radius-essential edges in graph \( G \). In this paper, we study realizability questions relating to the number of radius-essential edges, give bounds on \( c_r(G) \) in terms of radius and order, and we characterize various classes of graphs achieving extreme values of \( c_r(G) \).

We prove tight estimates on the minimum weight of an edge decomposition of the complete graph into subgraphs of 3 or 4 edges, where the weight of a subgraph is the number of its vertices. We conjecture that the weighted edge decomposition problem on general graphs is NP-complete for every \( k > 2 \). This conjecture is shown to be true for every \( k \leq 11 \) except \( k = 8 \). The problem is motivated by the traffic grooming problem for optical networks.

There are six distinct ways in which the vertices of a 4-cycle may be coloured with two colours, called \(\text{colouring types}\). Let \( C \) be the set of these colouring types and let \( S \) be a non-empty subset of \( C \). Suppose we colour the vertices of \( K_v \) with two colours. If \( D \) is a 4-cycle decomposition of \( K_v \) such that the colouring type of each 4-cycle is in \( S \), then \( D \) is said to have a \({colouring\; of\; type}\) \( S \). Furthermore, the colouring is said to be \({proper}\) if every colouring type in \( S \) is represented in \( D \). For all possible \( S \) of size one, two or three, excluding three cases already settled, we completely settle the existence question for 4-cycle decompositions of \( K_v \) with a colouring of type \( S \).

For a solution \( S \) of the \( n \)-queens problem, let \( M(S) \) denote the maximum of the absolute values of the diagonal numbers of \( S \), and let \( m(S) \) denote the minimum of those absolute values. For \( n \geq 4 \), let \( F(n) \) denote the minimum value of \( M(S) \), and let \( f(n) \) denote the maximum value of \( m(S) \), as \( S \) ranges over all solutions of the \( n \)-queens problem. Say that a solution \( S \) is an \( n \)-\({champion}\) if \( M(S) = F(n) \) and \( m(S) = f(n) \).

Approximately linear bounds are given for \( F(n) \) and \( f(n) \), along with computational results and several constructions together providing evidence that the bounds are excellent. It is shown that, in the range \( 4 \leq n \leq 24 \), \( n \)-champions exist except for \( n = 11, 16, 21, 22 \).

Let \( S \) be a stable set in a graph \( G \), possibly \( S = \emptyset \). The subgraph \( G – N[S] \), where \( N[S] \) is the closed neighborhood of \( S \), is called a \({co-stable \;subgraph}\) of \( G \). We denote by \( \text{CSub}(G) \) the set of all co-stable subgraphs of \( G \). A class of graphs \( \mathcal{P} \) is called \({co-hereditary}\) if \( G \in \mathcal{P} \) implies \( \text{CSub}(G) \subseteq \mathcal{P} \). Our result: If the set of all minimal forbidden co-stable subgraphs for a non-empty co-hereditary class \( \mathcal{P} \) is finite, then Stable Set is an NP-complete problem within \( \mathcal{P} \). Also, we prove that the decision problem of recognizing whether a graph has a fixed graph \( H \) as a co-stable subgraph is NP-complete for each non-trivial graph \( H \).

In this paper, we show the cordiality of the following families of graphs: (1) Pyramid graphs, (2) One point unions of plys,(3) One point unions of wheel related graphs, (4) Path unions of shells of different sizes, (5) Path unions of flags of different sizes.