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Let D be an acyclic orientation of a simple graph G. An arc is called dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define dmin(G) to be the minimum number of d(D) over all acyclic orientations D of G. Let Cn denote the cycle on n vertices. The cube Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. is the graph defined on the same vertex set of Cn such that any two distinct vertices u and v are adjacent in Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. if and only if their distance in Cn is at most 3. In this paper, we study the structure of Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. to determine its minimum number of dependent arcs. [ABSTRACT FROM AUTHOR]

Let ... be a connected digraph with n≥2 vertices. Suppose that a fire breaks out at a vertex v of .... A firefighter starts to protect vertices. At each time interval, the firefighter protects k vertices not yet on fire. Afterward, the fire spreads to all unprotected neighbors that are heads of some arcs starting from the vertices on fire. Let snk(v) denote the maximum number of vertices in ... that the firefighter can save when a fire breaks out at vertex v. The k-surviving rate ρk(...) of ... is defined as .... In this paper, we consider the k-surviving rate of digraphs. Main results are as follows: (1) if ... is a k-degenerate digraph, then ρk(...)≥1/k+1; (2) if ... is a planar digraph, then ρ2(...)>1/40; (3) if ... is a planar digraph without 4-cycles, then ρ1(...)>1/51. [ABSTRACT FROM AUTHOR]