Recurrent points and non-wandering points of graph maps

Let G be a graph and f : G → G be a continuous map. Denote by P ( f ) , R ( f ) and Ω ( f ) the sets of periodic points, recurrent points and non-wandering points of f, respectively. In this paper we show that: (1) If L = ( x , y ) is an open arc contained in an edge of G such that { f m ( x ) , f k ( y ) } ⊂ ( x , y ) for some m , k ∈ N , then R ( f ) ∩ ( x , y ) ≠ ∅ ; (2) Any isolated point of P ( f ) is also an isolated point of Ω ( f ) ; (3) If x ∈ Ω ( f ) − Ω ( f n ) for some n ∈ N , then x is an eventually periodic point. These generalize the corresponding results in W. Huang and X. Ye (2001) [9] and J. Xiong (1983, 1986) [17,19] on interval maps or tree maps.