Moving **Vertices** to Make Drawings Plane ⋆ Xavier Goao c 1, Jan Kratochv´õl 2, Yoshio Okamoto 3⋆⋆, Chan-Su Shin 4⋆⋆⋆, and Alexander Wolf f 5 1 LORIA-INRIALorraine, Nancy, France. goaoc@loria.fr 2 Dept. Applied Math. and Inst. Theoret.

Lesson 26 AIM: How do we prove general theorems, using coordinate geometry? PERFORMANCE STANDARDS: M2, M3, M6 PERFORMANCE OBJECTIVES: The student will be able to...

Problem 1: Prove that every simple graph with at least two **vertices** has at least two **vertices** of the same degree. Solution: Let G be a simple graph with V **vertices**.

A graph Gis Eulerianifand only if it has at most one nontrivial component and its **vertices** all have even degree. There are at least threedifierent approaches to the proof of this theorem.

(a) Explain how to model this trick as a bipartite matching problem between the 13 column ver tices and the 13 value **vertices**. Is the graph necessarily degree constrained?

feasible solutions in. Mathematical Programming 59 (1993) 23-31 23 North-Holland Determination of optimal **vertices** from feasible solutions in unimodular linear programming Shinji Mizuno Department of Prediction and Control, The Institute of Statistical Mathematics, Tokyo, Japan.. ..' .

On total domination and support **vertices** of a tree Ermelinda DeLaVi˜na †, Craig E. Larson ‡, Ryan Peppe r †and Bill Waller † † University of Houston-Downtown, Houston, Texas 77002 delavinae@uhd.edu, pepperr@uhd.edu, wallerw@uhd.edu ‡ Virginia Commonwealth University, Richmond ...

Theorem: Every graph has an even number of **vertices** with odd degree. Proof:

6.3 Trees A path in an undirected graph G = (V, E) is a sequence of edges that connect adjacent **vertices**. An undirected graph G = (V, E) is connected if there is a path between any pair of **vertices**.

Interpreting Line Drawings of Objects with K -**Vertices** P. A. C. Varley, H. Suzuki Department of Precision Engineering, The University of Tokyo, Tokyo, Japan {pvarley, suzuki}@cim.pe.u-tokyo. ac.jp R. R. Martin School of Computer Science, Cardiff University, Cardiff, Wales, UK ralph@cs.cf.ac.uk ...