By Kusahara T.

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**Extra info for A barrier method for quasilinear ordinary differential equations of the curvature type**

**Example text**

If G is a topological planar graph then G∗ is planar. 14). 10. 14. A planar graph and its dual Let us recall that a set of edges is even if it induces even degrees at each vertex. We will say that a subset E of edges of a plane graph G is dual even if {e∗ ; e ∈ E } is an even set of edges of G∗ . 6. The dual even subsets of edges of G are exactly the edge cuts of G. 7. Let G, G∗ be connected multigraphs and let f be a bijection between their edge-sets. We say that G∗ is a combinatorial dual of G if, for each F ⊂ E(G), F is the edge-set of a cycle of G if and only if f (F ) is a minimal (with respect to inclusion) edge cut in G∗ .

The symmetric diﬀerence of M and M consists of vertex-disjoint alternating paths and cycles. 36 CHAPTER 2. INTRODUCTION TO GRAPH THEORY Since M is bigger, there must be an alternating path that starts and ends by an edge out of M . Hence, we get the following. 13. M is maximum if and only if there is no alternating path with both endvertices not covered by M . Such a path will be called an augmenting path. Let M be a matching. An alternating trail W = (v0 , e0 , v1 , · · · , vt ) is called a ﬂower of M if t is odd, v0 , · · · , vt−1 are all distinct, v0 not covered by M and vt = vi for some even i, 0 ≤ i < t.

Clearly, there is a bijection between the set of the ﬂows of D and the set of the ﬂows of D and the s, t−cuts of D of ﬁnite capacity exactly correspond to the s, t−vertex cuts. 5 Connectivity Let G = (V, E) be a graph and let s, t be two distinct vertices of G. We say that two s, t−paths are independent if they have only the vertices s, t in common. The following Menger’s theorem is perhaps the most important theorem regarding graph connectivity. 1. Let s and t be distinct non-adjacent vertices of a graph G.