For a connected graph G = (V,E), a set S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining some pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is gev(G). Any edge-to-vertex geodetic set of cardinality gev(G) is called an edge-to-vertex geodetic basis of G. A subset T ⊆ S i...

Myers conjectured that for every integer s there exists a positive constant C such that for all integers t every graph of average degree at least Ct contains a Ks,t minor. We prove the following stronger result: for every 0 < ε < 10−16 there exists a number t0 = t0(ε) such that for all integers t ≥ t0 and s ≤ εt/ log t every graph of average degree at least (1 + ε)t contains a Ks,t minor. The b...

As a strengthening of the concept global forcing number graph G, complete G is cardinality minimum edge subset to which restriction every perfect matching M set M. Xu et al. (J Comb Opt 29: 803–814, 2015) revealed that also antifores each matching, and obtained for catacondensed hexagonal system, equal Clar plus hexagons (Chan MATCH Commun Math Comput Chem 74: 201–216, 2015). In this paper, we ...

The positive zero forcing number of a graph is a graph parameter that arises from a non-traditional type of graph colouring, and is related to a more conventional version of zero forcing. We establish a relation between the zero forcing and the fast-mixed searching, which implies some NP-completeness results for the zero forcing problem. For chordal graphs much is understood regarding the relat...

A subset S of initially infected vertices of a graph G is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects this neighbour. The forcing number of G is the minimum cardinality of a forcing set in G. In the present paper, we study the forcing number of various classes o...