9   About Intersection Graphs
Since this lecture we focus on selected detailed topics in Graph theory that are "close to your teacher's heart" ...
The first selected topic is that of intersection graphs, i.e. of graphs that are defined by the intersecting pairs of certain - often geometric - objects. This area of graphs is motivated both by its illustartive nature and by its practical applicability (e.g. interval graphs).
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Brief outline of this lecture
• What are intersection graphs; the interval graphs as an example.
• Chordal graphs and their properties.
• Several more commonly studied intersection and geometric classes.
• String and segment representations of graphs as another example.
9.1   Intersection graphs; Interval graphs
Definition 9.1. The intersection graph of a set family M
is the graph IM on the vertices V — M and edges E — {{A, B} cM : An B ^
We remark that typical examples of intersection graphs are of geometric nature, c
Fact: Specific intersection graph classes are always closed on induced subgraphs.
Proposition 9.2. Every simple graph is isomorphic to the intersection graph of a suitable set system.
Interval graphs
One of the oldest studied examples of intersection graphs are the interval graphs (shortly INT) - the intersection graphs of intervals on a line.
Recall that these graphs have already been implicitly used in connection with the single job assignment problem, which actually was the colouring problem for interval graphs.
Lemma 9.3. Every cycle of length more than three in an interval graph has a chord.
Theorem 9.4. The class of interval graphs has the following characterizations: □ • A graph is INT if and only if it has no induced subgraph isomorphic to one of
• A simple graph G is INT if and only ifG has no induced C4, and the complement of G has a transitive orientation.
9.2   Chordal graphs
Definition 9.5. Chordal graph G (also called triangulated)
is such a graph G with no induced cycle (i.e. no chordless cycle) of length > 3 in G. For example:
□
Theorem 9.6. Every chordal graph G contains a simplicial vertex, which is a vertex s such that the neighbours of s in G form a clique. □
Proof: We moreover say that a graph H is bisimplicial if H is complete, or H contains two nonadjacent simplicial vertices. Then.. .
The proof is acomplished in the following tricky sequence of three relatively straightforward claims:
1. It holds that for every cycle C in any chordal graph G, and an edge e there exists an edge / in G such that E(C) \ {e} U {/} contains a triangle, c
2. Let e — uv be an edge of G and let Na(v) - the neighbours of v - induce a bisimplicial subgr. of G. If v is simplicial in Na(u) but not in whole G, then there is another w adjacent to v but not to u, such that w is simplicial in Na(v).
Xq — u
3. Hence if G is not bisimplicial, but the neighbourhoods of its vertices all induce bisimplicial subgraphs, then G contains a cycle C contradicting (1).
4. Therefore, G is bisimplicial. C
Simplicial decomposition
From Theorem 9.6, one can easily conclude:
Corollary 9.7. Every chordal graphs has a simplicial decomposition, i.e. a vertex ordering V(G) — (t>i, t>2, • • •, vn) such that each Vi, i — 2,..., n, is simplicial in the subgraph induced on the vertex subset {v\,..., fi-i}.
Fact: Simplicial decompositions can be used to build efficient recognition algorithms for chordal and interval graphs.
Another intersection characterization
The following shows that chordal graphs actually present a natural generalization of interval graphs...
Theorem 9.8. A graph G is chordal if and only if there exists a tree T such that G is the intersection graph of a collection of subtrees in T. □
Proof (only a sketch of =4> ); by induction on the number of vertices of G:
• This is trivial for one vertex, c
• Let v be a simplicial vertex of G, and let Go — G—v. Then Go has an intersection representation by subtrees in a tree To. c
The neighbours of v in G form a clique K C G, and all the trees representing vertices of K must intersect in a joint node x G V(Tq). We construct T by adding a new leaf y in To adjacent to x, and represent the vertex v by a tree {y}.
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9.3   More intersection graph examples
We briefly and informally introduce few more commonly studied types (classes) of intersection graphs, mostly of geometric nature.
• A line graph L(G) of a graph G is the intersection graph of the edges E(G).
• Circular-interval graphs (CA) are the intersection graphs of intervals on a circle.
□
• Circle graphs (CIR) are the intersection graphs of straight chords of a circle.
Pa C5 Pa
• Disc graphs (DISC) are the intersection graphs of closed discs in the plane. Furthermore, unit-disc graphs are such that all the discs have unit size.
Notice that these classes can be considered as generalizations of interval graphs.
Petr Hliněný,  Fl MU Brno, 2013
10/17
Fl: MA010: Intersection Graphs
Complexity of recognition
Theorem 9.9. The class recognition problem - to decide whether a given abstract graph belongs to the specified class, is
• polynomial time solvable for INT, line graphs, CA, and CIR; □
• and NP-hard for DISC, unit-DISC, BOX (in any dimension >2).c
Theorem 9.10. A graph G is a line graph of a simple graph if, and only if, G does not contain any of the following induced subgraphs:
Petr Hliněný,  Fl MU Brno, 2013
11/17
Fl: MA010: Intersection Graphs
Contact (touching) graphs
Considering intersection graphs of geometric objects, it is sometimes natural to define the following restriction:
• Contact graphs are the graphs having an intersection representation such that the objects "do not overlap" (formally, their topol. interiors are pairwise disjoint).
A particularly beautiful result of Koebe reads:
Theorem 9.11. A graph G is planar if, and only if, G is a contact graph of discs in the plane (a coin graph).
Petr Hliněný,   Fl MU Brno,   2013 12/17 Fl: MA010: Intersection Graphs
• Rectangular duals - another example: Those are the graphs having a contact representation by a collection of (non-overlap.) axis-parallel rectangles in the plane.
- Note; meeting of four rectangles in one point is disallowed! c
Fact: Only planar graphs can have rectangular duals, but not all planar ones have.
- In a strict sense, rectangual dual repres. must "fill the plane without holes".
Fact: Strict rectangular duals always represent planar quasi-triangulations (all faces except the outer one are triangles).
Petr Hliněný,  Fl MU Brno, 2013
13/17
Fl: MA010: Intersection Graphs
9.4   Curve and line segment intersection graphs
• String graphs are the intersection graphs of simple curves in the plane. For example, every planar graph is a string graph, see:
□
On the other hand, not all graphs are string graphs (the smallest non-string graphs have 12 vertices.) □
Moreover, the structure of string representations can be very complicated.
Proposition 9.13. There exist string graphs such that every their representation contains a pair of curves having exponentially many intersections.
Petr Hliněný,  Fl MU Brno, 2013
14/17
Fl: MA010: Intersection Graphs
• Segment intersection graphs are the intersection graphs of straight line segments in the plane.
For example, see: ._._
This is a proper subclass of string graphs, and the structure is again quite complicated. For instance, there exist segment intersection graphs such that every their representation requires double-exponential precision of segment coordinates, c
Theorem 9.14. The class recognition problems are NP-hard for both string and segment intersection graphs.
□
Petr Hliněný,  Fl MU Brno, 2013
15/17
Fl: MA010: Intersection Graphs
Some more difficult findings
On the other hand, the following two statements are highly nontrivial and their proofs have been searched for many years.
Theorem 9.15. The recognition problem of string graphs is in NP. [Schaeffer and Stefankovic]
Recall that a string intersection representation may be exponentially complex. . . □
Theorem 9.16. Every planar graph is a segment intersection graph. [Chalopin, Goncalves] c
Question. How many "segment slopes" one needs to represent every planar graph as a segment intersection graph?
Petr Hliněný,  Fl MU Brno, 2013
16/17
Fl: MA010: Intersection Graphs
"Match" graphs
Similarly to coin graphs, one can straightforwardly define a special subclass of segment graphs in which the line segments in an intersection representation are not allowed to cross in their interior points (i.e., having pairwise disjoint interiors).
• The aforementioned class is called the class of segment contact graphs, c
Theorem 9.17. A graph G is a segment contact graph of only vertical and horizontal segments if, and only if, G is a planar bipartite graph.
Though.. .
Theorem 9.18. The recognition problem is NP-complete for (general) segment contact graphs.
Petr Hliněný,  Fl MU Brno, 2013
17/17
Fl: MA010: Intersection Graphs