The decycling number and maximum genus of cubic graphs

Abstract

Let ∇(G) and γMfalse(Gfalse) denote the minimum size of a decycling set and maximum genus of a graph G, respectively. For a connected cubic graph G of order n, it is shown that ∇false(Gfalse)+γMfalse(Gfalse)=n2+1. Applying the formula, we obtain some new results on the decycling number and maximum genus of cubic graphs. Furthermore, it is shown that the number of vertices of a decycling set S in a k‐regular graph G is false|Sfalse|=1k−1false{β(G)+c+mS′−1false}, where c and mS′ are the number of components of G−S and the number of edges in G[S], respectively. Therefore, S is minimum if and only if c+mS′−1 is minimum. As an application, this leads to a lower bound ∇false(Gfalse)≥false⌈βfalse(Gfalse)k−1false⌉ for ∇(G) of a k‐regular graph G. In many cases this bound may be sharp.

Nonseparating Independent Sets of Cartesian Product Graphs

“…Let G be a connected graph with a Xuong-tree T X . Then there exists an edge-partition of G − E(T X ) as follows: (2) Huang and Liu [5], and Ren and Long [7], respectively, proved that Z(G) = γ M (G) holds for each cubic graph G.…”

Feedback Arc Number and Feedback Vertex Number of Cartesian Product of Directed Cycles

“…e readers are referred to [5,6] for a review of some earlier results and open problems, and [7][8][9] for some recent results on the feedback vertex number of graphs. Some bounds or exact values are established for various families of graph, for instance, outerplanar graphs [10], grids and butter ies [11], cubic graphs [12,13], bipartite graphs [14], generalized Petersen graphs [15], regular graphs [16,17]. Bau et al [18] investigated the feedback number of grid graphs.…”