Week 11 – Cop number of outerplanar graphs Dr. Anthony Bonato Ryerson University AM8002 Fall 2014

Planar graphs a graph is planar if it can be drawn in the plane without edge crossings Theorem 11.1 (Aigner, Fromme,84): planar graphs have cop number at most 3. 2

Example: Buckeyball graphs girth 5, 3-regular, so have cop number ≥ 3 (and so = 3 by theorem). 3

Ideas behind proof cop territory: induced subgraph so that if the robber entered he would eventually be caught (not necessarily immediately) cop territory starts as a maximum order isometric path inductively grow cop territory, until it is entire graph show that the cop territory is always one of three kinds, and that we can always enlarge it so it remains one of the three kinds 4

Outerplanar graphs a graph is outerplanar if its vertices can be arranged on a circle with the following properties: 1.Every edge joins two consecutive vertices on the circle, or forms a chord on the circle. 2.If two chords intersect, then they do so at a vertex. 5

Examples cycles each of these are maximal outerplanar: 6

Characterization Theorem 11.2 (Kuratowski,1930) A graph is planar if it does not contain a subdivision of K 5 or K 3,3. Theorem 11.3 (Harary, Chartrand,1967) A graph is outerplanar if it does not contain a subdivision of K 4 or K 2,3. 7

Cop number of outerplanar graphs Theorem 11.4 (Clarke, 02) If G is outerplanar, then it’s cop number is at most 2. proof is simpler than planar case, but still needs some care overall idea is similar: enlarge cop territory two cases: no cut vertices, or some cut vertices 8