The time complexity of a search depends on (1) the size of the neighbourhood and (2) the complexity in determining the cost of the moves. Therefore in order for the search to be effective and efficient, the neighbourhood must be highly constrained and it needs to be evaluated quickly, however without losing the ability to provide access to improved solutions. This paper investigates strategies ...

Consider a configuration of pebbles distributed on the vertices of a connected graph of order n. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles on a graph is called solvable if it is possible to place a pebble on any given vertex using a sequence of pebbling steps. The pebbling number of a graph, denot...

A harmonious colouring of a simple graph G is a colouring of the vertices such that adjacent vertices receive distinct colours and each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. We improve an upper bound on h(G) due to Lee and Mitchem, and give upper bounds for related quantities.

Fukuda and Handa [5] asked whether every even partial cube G is harmonic. It is shown that the answer is positive if the isometric dimension of G equals its diameter. This is in particular true if G has isometric dimension at most 6. Under an additional technical condition it is proved that an even G is harmonic or has two adjacent vertices whose diametrical vertices are at distance at least 4....

An additive labeling of a graph G is a function l : V (G) → N, such that for every two adjacent vertices v and u of G, ∑

A pebbling move on a graph consists of taking two pebbles off of one vertex and placing one pebble on an adjacent vertex. In the traditional pebbling problem we try to reach a specified vertex of the graph by a sequence of pebbling moves. In this paper we investigate the case when every vertex of the graph must end up with at least one pebble after a series of pebbling moves. The cover pebbling...

Two vertices u and v in a nontrivial connected graph G are twins if u and v have the same neighbors in V (G) − {u, v}. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer k, let c : V (G) → Zk be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex c...

This paper examines the role that community generated information plays in ‘neighbourhood policing’ — a key component of the UK police reform agenda. The neighbourhood policing agenda is concerned with the delivery of a consistent presence of dedicated neighbourhood teams which should be visible and accessible to the community. However, it also calls for the generation of community intelligence...