Fractional Gallai–Edmonds decomposition and maximal graphs on fractional matching number

A fractional matching of a graph G is a function f that assigns to each edge a number in [0, 1] such that for each vertex v, $$\sum \nolimits _{e\in \Gamma (v)}f(e) \le 1$$∑e∈Γ(v)f(e)≤1, where $$\Gamma (v)$$Γ(v) is the set of all edges incident with v. The fractional matching number $$\mu _{f}(G)$$μf(G) of G is the supremum of $$\sum \nolimits _{e\in E(G)}f(e)$$∑e∈E(G)f(e) over all fractional matchings f of G. Let $$D_f(G)$$Df(G) be the set of vertices which are unsaturated by some maximum fractional matching of G, $$A_f(G)$$Af(G) the set of vertices in $$V(G)-D_f(G)$$V(G)-Df(G) adjacent to a vertex in $$D_f(G)$$Df(G) and $$C_f(G)=V(G)-A_f(G)-D_f(G)$$Cf(G)=V(G)-Af(G)-Df(G). In this paper, the partition $$(C_f(G), A_f(G), D_f(G))$$(Cf(G),Af(G),Df(G)), named fractional Gallai–Edmonds decomposition, is obtained by an algorithm in polynomial time via the Gallai–Edmonds decomposition. A graph G is maximal on $$\mu _{f}(G)$$μf(G) if any addition of edge increases the fractional matching number $$\mu _{f}(G)$$μf(G). The Turán number is the maximum of edge numbers of maximal graphs and the saturation number is the minimum of edge numbers of maximal graphs. In this paper, the maximal graphs are characterized by using the fractional Gallai–Edmonds decomposition. Thus the Turán number, saturation number and extremal graphs are obtained.