Θεωρία Γραφημάτων Θεμελιώσεις-Αλγόριθμοι-Εφαρμογές Ενότητα 8 Τ ΕΛΕΙΑ Γ ΡΑΦΗΜΑΤΑ Σταύρος Δ. Νικολόπουλος 1

2 Εισαγωγή Βασικοί Αλγόριθμοι Γραφημάτων Πολυπλοκότητα χώρου και χρόνου: Ο και Ω Τέλεια Γραφήματα Κλάσεις Ιδιότητες Προβλήματα Αλγοριθμικές Τεχνικές … Αλγόριθμοι Προβλημάτων Αναγνώρισης και Βελτιστοποίησης

3 Αλγόριθμοι Θεωρίας Γραφημάτων Πολυωνυμικοί Αλγόριθμοι… (Γραμμικοί) Προβλήματα: NP-Πλήρη Επιλογές Προσέγγιση Λύσης Περιορισμοί Ιδιοτήτων Τέλεια Γραφήματα, …

4 Κλάσεις Τέλειων Γραφημάτων

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6 Μοντελοποίηση Προβλήματος →

Let F be a family of nonempty sets. The intersection graph of F is obtained be representing each set in F by a vertex: x  y  S X ∩ S Y ≠  7 Intersection Graphs

The intersection graph of a family of intervals on a linearly ordered set (like the real line) is called an interval graph 2 I 1 I 3 I I 2 I 4 I I 7 5 unit internal graph proper internal graph - no internal property contains another 8 7 Intersection Graphs (Interval)

Circular-arc graphs properly contain the internal graphs. proper circular - arc graphs Intersection Graphs (Circular-arc)

A permutation diagram consists of n points on each of two parallel lines and n straight line segments matching the points. Π= [ 4,1, 3, 2 ] G[ Π ] 10 Intersection Graphs (Permutation)

11 Intersection Graphs (Permutation)

Intersecting chords of a circle Proposition. An induced subgraph of an interval graph is an interval graph. 12 Intersection Graphs (Chords-of-circle)

Triangulated Graph Property every simple cycle of length l > 3 possesses a chord Triangulated graphs (or chord graphs) 13 Triangulated Property

Transitive Orientation Property Each edge can be assigned a one-way direction in such a way that the resulting oriented graph (V, F): ab  F and bc  F  ac  F (  a, b, c  V) Graphs which satisfy the transitive orientation property are called comparability graph. 14 Transitive Orientation Property

ab  F and bc  F  ac  F (  a, b, c  V) 15 Transitive Orientation Property

Clique number ω(G) the number of vertices in a maximum clique of G 16 Stability number α(G) the number of vertices in a stable set of max cardinality  Max κλίκα του G ω(G) = 4 Max stable set of G α(G) = 3 a b c d f e b c d e a c f Graph Theoretic Foundations (1)

A clique cover of size k is a partition V = C 1 + C 2 +…+ C k such that C i is a clique. A proper coloring of size c (proper c-coloring) is a partition V = X 1 + X 2 +…+ X c such that X i is a stable set. 17 Graph Theoretic Foundations (2)

Clique cover number κ(G) the size of the smallest possible clique cover of G Chromatic number χ(G) the smallest possible c for which there exists a proper c-coloring of G. χ(G) = 2 κ(G) = 3 clique cover V={2,5}+{3,4}+{1} c-coloring V={1,3,5}+{2,4} κ(G)=3 χ(G)=2 18 Graph Theoretic Foundations (3)

For any graph G: ω(G) ≤ χ(G) α(G) ≤ κ(G) Obriasly : α(G) = ω(Ğ) and κ(G) = χ(Ğ) 19 Graph Theoretic Foundations (4)

Let G = (V, E) be an undirected graph: (P 1 ) ω(G A ) = χ(G A )  A  V (P 2 ) α(G A ) = κ(G A )  A  V G is called Perfect 20 Perfect Graphs - Definition

χ-Perfect property For each induced subgraph G A of G χ(G A ) = ω(G A ) α-Perfect property For each induced subgraph G A of G α(G A ) = κ(G A ) 21 Perfect Graphs - Properties

22 Κλάσεις Τέλειων Γραφημάτων

23 Κλάσεις Τέλειων Γραφημάτων

G triangulated  G has the triangulated graph property Every simple cycle of length l > 3 possesses a chord. Triangulated graphs, or Chordal graphs, or Perfect Elimination graphs Triangulated Graphs

Dirac showed that: every chordal graph has a simplicial node, a node all of whose neighbors form a clique. 25 a b c d f e a, c, f simplicial nodes b, d, e non siplicial Triangulated Graphs

It follows easily from the triangulated property that deleting nodes of a chordal graph yields another chordal graph. 26 Triangulated Graphs a b c d f e a b c d f a b d f  

Recognition Algorithm This observation leads to the following easy and simple recognition algorithm : Find a simplicial node of G Delete it from G, resulting G ’ Recourse on the resulting graph G ’, until no node remain 27 Triangulated Graphs

node-ordering : perfect elimination ordering, or perfect elimination scheme Rose establishes a connection between chordal graphs and symmetric linear systems. 28 a b c d e (a, c, b, e, d) (c, d, e, a, b) (c, a, b, d, e) … Triangulated Graphs

Let σ =  v 1,v 2,...,v n  be an ordering of the vertices of a graph G = (V, E). σ = peo if each v i is a simplicial node to graph G  v i,v i+1, …,v n  29 Triangulated Graphs a b c d e σ = (c, d, e, a, b) a b d e

Example: σ =  1, 7, 2, 6, 3, 5, 4  no simplicial vertex G 1 has 96 different peo. 30 Triangulated Graphs

Algorithm LexBFS 1. for all v  V do label(v) :  () ; 2. for i :   V  down to 1 do 2.1 choose v  V with lexmax label (v); 2.1 σ (i)  v; 2.3 for all u  V ∩N(v) do label (u)  label (u) || i 2.4 V  V \  v  ; end 31 Algorithms for computing peo

32 Algorithms for Computing PEO

Algorithm MCS 1. for i :   V  down to 1 do 1.1 choose v  V with max number of numbered neighbours; 1.2 number v by i; 1.3 σ (i)  v; 1.4 V  V \  v  ; end 33 Algorithms for Computing PEO

34 Algorithms for Computing PEO

35 Minimum Vertex Separators x y p q r z

36 Minimum Vertex Separators

37 Minimum Vertex Separators

Βασικοί Αλγόριθμοι Γραφημάτων Πολυπλοκότητα χώρου και χρόνου: Ο και Ω Τέλεια Γραφήματα Κλάσεις Ιδιότητες Προβλήματα Τεχνικές Διάσπασης (modular decomposition, …) Αλγόριθμοι Προβλημάτων Αναγνώρισης και Βελτιστοποίησης 38 Coloring Max Clique Max Stable Set Clique Cover Matching Hamiltonian Path Hamiltonian Cycle … Perfect Graphs – Optimization Problems Triangulated Comparability Interval Permutation Split Cographs Threshold graphs QT graphs …

Minimum Vertex Separators