Maximum cardinality neighbourly sets in quadrilateral free graphs

Abstract

Neighbourly set of a graph is a subset of edges which either share an end point or are joined by an edge of that graph. The maximum cardinality neighbourly set problem is known to be NP-complete for general graphs. Mahdian (Discret Appl Math 118:239–248, 2002) proved that it is in polynomial time for quadrilateral-free graphs and proposed an $O(n^{11})$ algorithm for the same, here n is the number of vertices in the graph, (along with a note that by a straightforward but lengthy argument it can be proved to be solvable in $O(n^5)$ running time). In this paper we propose an $O(n^2)$ time algorithm for finding a maximum cardinality neighbourly set in a quadrilateral-free graph.

Appendix

Mahdian states and proves the following theorem:

Theorem 2

(Mahdian (2002)) In a quadrilateral-free graph G, for any neighbourly set A at least one of the following conditions hold:

• 1. There exists a triangle $\triangle uvw$ such that $A \subseteq E_u \cup E_v \cup E_w$

• 2. There exists an edge uv such that $A \subseteq E_u \cup E_v$

• 3. There exists a vertex u such that A has at most 10 edges not in $E_u$

In Mahdian (2002) the author adds a remark, that the constant ten in Theorem 2 can be replaced by 4 and claims that would lead to an $O(n m^4)$ time algorithm. It appears that the claim in the remark may not be correct (at least is not that obvious) as can be seen from the following example (see Fig. 20).

Fig. 20

Counter example for remark in Mahdian (2002)

Let neighbourly set A consists of three disjoint paths abc,def and ghi of length two (each) and an additional edge lm, vertex disjoint, with these paths.

The connecting non-neighbourly set edges in the graph are be, bh, eh (as in Fig. 19) and lm is connected with edges

$$\begin{aligned} al, cm, dl, fm, gl, im \end{aligned}$$

to each path of length two (as in Fig. 16b).

This graph is clearly quadrilateral free and A is neither in form (1) nor in form (2) of Theorem 2.

As each vertex has a maximum degree of two in A, and as there are seven edges in A, there is no vertex u for which A has 4 edges not in $E_u$ (for example, five edges de, ef, hg, hi, lm are not incident to b).