Descriptive Complexity of Graph Classes via Labeling Schemes Maurice Chandoo FernUniversität in Hagen ACTO Seminar University of Liverpool May 2021 2 7 1 3 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng adjacency matrix 2(1+log n) 0 0 ··· 1 1 labeling `: V(G) ! f0; 1g . @ . .. A 1 ··· 0 requires n2 bits u `(u) 0 fu; vg 2= E(G) AIntv alternative representation v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? interval graph G Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng adjacency matrix 2(1+log n) 0 0 ··· 1 1 labeling `: V(G) ! f0; 1g . @ . .. A 1 ··· 0 requires n2 bits u `(u) 0 fu; vg 2= E(G) AIntv alternative representation v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? interval graph G Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng 2(1+log n) 0 0 ··· 1 1 labeling `: V(G) ! f0; 1g . @ . .. A 1 ··· 0 requires n2 bits u `(u) 0 fu; vg 2= E(G) AIntv alternative representation v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? interval graph G adjacency matrix Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng 2(1+log n) 0 0 ··· 1 1 labeling `: V(G) ! f0; 1g . @ . .. A 1 ··· 0 requires n2 bits u `(u) 0 fu; vg 2= E(G) AIntv alternative representation v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 1 2 3 4 5 interval graph G adjacency matrix Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) requires n2 bits u `(u) 0 fu; vg 2= E(G) AIntv alternative representation v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 1 2 3 4 5 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv alternative representation v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 1 2 3 4 5 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 1 2 3 4 5 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits alternative representation Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits alternative representation Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 3 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 1 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits alternative representation Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 7 3 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 2 1 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits alternative representation Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 7 4 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 2 1 3 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits alternative representation Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 7 5 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 2 1 3 4 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits alternative representation Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 7 6 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 2 1 3 4 5 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits alternative representation Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 7 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 2 1 3 4 5 6 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits alternative representation Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 8 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 2 7 1 3 4 5 6 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits alternative representation Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 9 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 2 7 1 3 4 5 6 8 interval graph G adjacency matrix 0 0 ··· 1 1 . @ . .. A 1 ··· 0 requires n2 bits alternative representation Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 10 label `(v) 2 f1;:::; 2ng × f1;:::; 2ng labeling `: V(G) ! f0; 1g2(1+log n) u `(u) 0 fu; vg 2= E(G) AIntv v `(v) fu; vg 2 E(G) f(1; 3); (2; 7); (4; 5);::: g 1 requires n ·O(log n) bits label decoding algorithm What is a Labeling Scheme? 2 7 1 3 4 5 6 8 9 interval graph G adjacency matrix 0 0 ··· 1 1 .