A mathematics project of Prof. Peter Borg funded by SEA-EU grant 2025

In 2025, Prof. Peter Borg was one of the three recipients of the SEA-EU grant award by the University of Malta (UM).

His grant funded a mathematics research project carried out in collaboration with academics from University of Gdańsk (UG) and Gdańsk University of Technology (GUT). He made a research visit at UG during 14 - 28 June 2025 for this purpose. He was joined by his two doctoral students, Mr Karl Bartolo and Mr Dayle Scicluna, who were both supported by an Erasmus grant.

The project led by Prof. Borg yielded a research paper that is published online. His co-authors are Mr Karl Bartolo (UM), Dr Magda Dettlaff (UG), Dr Magdalena Lemańska (GUT) and Prof. Paweł Żyliński (UG).

The paper was also submitted to a reputable peer-reviewed journal and is under review. It contains new results in graph theory, which is the mathematics of relations and connections, and has numerous real-life applications, ranging from all forms of networks (social, computer, internet, transport) to chemistry.

In graph theory, a graph G is not the usual plot but essentially a network, consisting of a set of objects, called vertices, together with a set of relations, called edges.

If {x, y} is an edge of G, then this signifies that the vertices x and y of G are related. For a subset S of the set V(G) of vertices of G, the closed neighbourhood N[S] consists of the vertices in S and each vertex related to at least one of them. If N[S] = V(G), then S is called a dominating set of G.

Domination theory is the popular study of dominating sets. In a seminal paper (Partial domination - the isolation number of a graph, Filomat 31(12) (2017), 3925-3944), Professors Yair Caro (University of Haifa-Oranim, Israel) and Adriana Hansberg (National Autonomous University of Mexico, Mexico) widened the study of dominating sets to the study of isolating sets.

As was the case of domination, the study of isolation has taken graph theory by storm and is now one of its most active fields of investigation. The most natural case of the vast isolation problem beyond the domination problem is that of determining how small a subset S of V(G) can be if each edge of G has at least one vertex in N[S]. Such a set S is called an isolating set of G. The size (number of members) of a smallest one is called the isolation number of G.

Edge subdivision is a graph operation that has various applications. It involves ‘inserting a vertex on an edge’. More precisely, it replaces an edge {x, y} by two edges {x, z} and {z, y}, where z is a new vertex (added only for x and y). The paper investigates how subdividing edges affects the isolation number. It determines the graphs whose isolation number increases upon subdividing any edge. It also shows that subdividing all edges, or all but one, increases the isolation number, and that this result is best possible.