It has been conjectured that the smallest cardinality \(\theta(G)\) of a perfect neighbourhood set of a graph is bounded above by ir\((G)\), the smallest order of a maximal irredundant set.
We prove results concerning the construction of perfect neighbourhood sets from irredundant sets which could help to resolve the conjecture and which establish that \(\theta(G) \leq \text{ir}(G)\) in certain cases.
In particular, the inequality is proved for claw-free graphs and for any graph which has an ir-set \(S\) whose induced subgraph has at most six non-isolated vertices.

We introduce and study two new parameters, namely the upper harmonious chromatic number, \(H(G)\), and the upper line-distinguishing chromatic number, \(H'(G)\), of a graph \(G\).
\(H(G)\) is defined as the maximum cardinality of a minimal harmonious coloring of a graph \(G\), while \(H'(G)\) is defined as the maximum cardinality of a minimal line-distinguishing coloring of a graph \(G\).
We show that the decision problems corresponding to the computation of the upper line-distinguishing and upper harmonious chromatic numbers are NP-complete for general graphs \(G\).
We then determine \(H'(P_n)\) and \(H(P_n)\).
Lastly, we show that \(H\) and \(H’\) are incomparable, even for trees.

For a graph \(G\), assign an integer value weight to each edge. For a vertex \(v\), the label of v is the sum of weights of the edges incident with it. Further, the weighting is irregular if all the vertex labels are distinct. It is well known that if \(G\) has at most one isolated vertex and no isolated edges, then there exist irregular assignments, in fact, using positive edge weights.

In this paper, we consider the following special weighting:
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– If \(G\) has order \( 2 k + 1\), then a consecutive labeling is an assignment where the vertex labels are precisely \(-k, -k+1, \ldots, -1, 0, 1, 2, \ldots, k-1, k\).

– If \(G\) has order \( 2k\), then a consecutive labeling is an assignment where the vertex labels are precisely \( -k+1, \ldots, -1, 0, 0, 1, 2, \ldots, k-1\).
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Here we show that every graph which has an irregular assignment also has a consecutive labeling. This concept is extended by considering all consecutive labelings and looking for one that has the smallest maximum, in absolute value, edge weight. This weight is referred to as the consecutive strength. Results parallel to the concept of irregularity strength are presented.

A modification of the Schreier-Sims algorithm is described which builds a permutation group utilising the transitivity of the stabiliser subgroups. Alternating and symmetric groups are recognised by their transitivity, resulting in a greatly improved time to build symmetric and alternating groups.
The algorithm has applications to graph isomorphism and other combinatorial isomorphism algorithms, as well as permutation group algorithms.

Suppose \(G = (V, E)\) is a graph in which every vertex \(v\) has a non-negative real number \(\omega(v)\) as its weight. The \(\omega\)-distance sum of \(v\) is \(D_{G,\omega}(v) = \sum_{u \in V} d(v, u)\omega(u).\) The \(\omega\)-median \(M_\omega(G)\) of \(G\) is the set of all vertices \(v\) with minimum \(\omega\)-distance sum \(D_{G,\omega}(v)\). This paper gives linear-time algorithms for computing the \(\omega\)-medians of interval graphs and block graphs.

Let \(p\) denote the number of vertices in a graph and let \(q\) denote the number of edges. Two cycles in a graph are disjoint if they have no common vertices. Pósa proved that any graph with \(q \geq 3p – 5\) contains two disjoint cycles. This result does not apply to planar graphs, since every planar graph has \(q \leq 3p – 6\).
In this paper, I show that any planar graph with \(q \geq 2p\) contains two disjoint cycles. I also show that this bound is best possible and that there is no minimum number of edges in a planar graph which will ensure the graph contains \(3\) disjoint cycles. Furthermore, a sufficient condition for any triangle-free graph (and therefore any bipartite graph) to contain \(k\) disjoint cycles is given.

The domination graph of a digraph is the graph on the same vertices with an edge between two vertices if every other vertex loses to at least one of the two. This note describes which connected graphs are domination graphs of tournaments.

Sharp invariant relationships involving various types of domination numbers are found
between a graph and its line graph.

A well-known problem in domination theory is the long-standing conjecture of V.G. Vizing from 1963 (see [7]) that the domination number of the Cartesian product of two graphs is at least as large as the product of the domination numbers of the individual graphs.
Although limited progress has been made, this problem essentially remains open. The usefulness of a maximum 2-packing in one of the graphs in establishing a lower bound has been recognized for some time.
In this paper, we shall extend this approach so as to take advantage of 2-packings whose membership can be altered in a certain way. This results in an improved lower bound for graphs which have 2-packings of this type.

A graph \(G\) is claw-free if it does not contain any complete bipartite graph \(K_{1,3}\) as an induced subgraph, and closed claw-free if it is the line-graph of a triangle-free graph. The inflation \(H_1\) of a graph \(H\) is obtained from \(\mathop{H}\limits^{i}\) by replacing each vertex \(x\) of degree \(d(x)\) by a clique \(X \simeq K_{d(x)}\).
Every inflated graph \(G = H_1\) is closed claw-free.
The minimum cardinalities \(\gamma(G)\), \(\text{ir}(G)\), and \(\text{rai}(G)\) of respectively a dominating set, a maximal irredundant set, and an \(R\)-annihilated irredundant set of any graph \(G\) satisfy
\(\text{rai}(G) \leq \text{ir}(G) \leq \gamma(G).\)
The motivation of this paper is that for inflated graphs, it is known that the difference \(\gamma(G) – \text{ir}(G)\) can be arbitrarily large, but not how large the ratio \(\gamma(G)/\text{ir}(G)\) can be. We show that \(\gamma(G) \leq 3\text{rai}(G)/2\) for every claw-free graph \(G\) and study the sharpness of the bounds
\(1 \leq {\gamma(G)}/{\text{ir}(G)} \leq {\gamma(G)}/{\text{rai}(G)} \leq {3}/{2}\)
in the four classes of claw-free graphs, closed claw-free graphs, inflated graphs, and line graphs of bipartite graphs.