AbstractThe interior of an orthogonal polygon drawn on a regular grid of the plane defines a set of cells (or squares) called a polyomino. We prove that the intersection graph of the maximal rectangles contained in a polyomino is slightly triangulated or has a star cutset

Maire, Frédéric

Elsevier - Publisher Connector 

A characterization of intersection graphs of the maximal rectangles of a polyomino 

Discrete Mathematics 120 (1993) 211-214 
North-Holland 
211 
Communication 
A characterization of intersection 
graphs of the maximal rectangles 
of a polyomino 
Frkdkric Maire 
Universith Paris 6, UFR 921, Equipe Combinatoire, 4 Place Jussieu, 75005 Paris, France 
Communicated by C. Berge 
Received 2 February 1993 
Abstract 
The interior of an orthogonal polygon drawn on a regular grid of the plane defines a set of cells (or 
squares) called a polyomino. We prove that the intersection graph of the maximal rectangles 
contained in a polyomino is slightly triangulated or has a star cutset. 
1. Introduction 
Shearer proved in [l] that the intersection graph of the maximal rectangles of 
a polyomino is perfect. This problem was posed in [3]. It seems difficult to deduce 
a characterization for this class of graphs with Shearer’s proof. We would like to find 
simple reasons why these graphs are perfect. Here we present a new proof of Shearer’s 
theorem, which uses a recent result on minimal imperfect graphs [2]. 
As usual in Perfect Graph Theory, the subgraphs we are interested in are those 
which are induced by a subset of vertices. For an introduction to perfect graph Theory 
see [4] or [S]. We write HE G to represent the fact that H is an induced subgraph of 
G. Let Pk denote the chordless path with k vertices and Ck the chordless cycle with 
k vertices. Let c denote the complement of the graph G. We say that a vertex x is loose 
if its neighbourhood is P,-free. In [2] we proved the following. 
Correspondence to: FrBdBric Maire, Universitk Paris 6, UFR 921, Equipe Combinatoire, 4 Place Jussieu, 
75005 Paris, France. 
0012-365X/93/$06.00 0 1993-Elsevier Science Publishers B.V. All rights reserved 
212 F. Maire 
Lemma 1. A minimal imperfect graph G without chordless cycle of length 5 or more has 
no loose vertex. 
A graph is called slightly triangulated if it contains no chordless cycle with 5 or more 
vertices, and if every induced subgraph has a loose vertex. The lemma above implies 
that these graphs are perfect. They generalize triangulated graphs. We first consider 
the cycles of the intersection graph G(P) of the maximal rectangles of a polyomino P, 
and show that the length of a chordless cycle in G(P) is less than 5. Then we establish 
that every induced subgraph of G has a loose vertex or a star cutset. At this point we 
have the following. 
Theorem 1. The intersection graph G(P) of the maximal rectangles of a polyomino P has 
no chordless cycle of length 5 or more, and every induced subgraph contains a loose 
vertex or has a star cutset. 
Hence G(P) is perfkt. In the following G stands for the intersection graph G(P) of 
the maximal rectangles of a polyomino P. The graph G is derived from P as follows. 
Let the vertices of G be the maxima1 rectangles contained in P. Let two vertices of G be 
joined by an edge iff the rectangles have a non-empty intersection. 
2. Cycles of G(P) 
A (geometric) vertex of a maximal rectangle of P will be called a corner to avoid 
confusion with the vertices of the graph. The interior of a rectangle is the interior in the 
usual Euclidean topological sense. Let x and y be two maximal rectangles of P. Since 
the rectangles are maximal we have the following. 
Proposition 1. A corner of x is in the interior of y if and only if a corner of y is in the 
interior of x. 
The intersection of x and y is called a cross if no corner of a rectangle is contained in 
the interior of the other. Otherwise the intersection is called a step. 
Let C=(rl,..., r,J, k>5, be a chordless cycle of G. Let ai be a cell of rinri+r. 
We note Wi,j the polygonal line from ai to aj through the segments 
[ai,ai+11,Cai+l,ai+21,..., [aj_ I, aj]. We are counting modulo k. We assume that C is 
chosen with a minimum number of step intersections. The smallest simply connected 
set containing C is an orthogonal polygon for which the rectangles of C are maximal 
too. We can thus assume that P is this polyomino. Suppose that there is an i such that 
rinri+r is a step. Then ri-Inri and ri+lnri+2 are crosses. Otherwise, if for example 
rim 1 n ri is a step then one of the rectangles of the sequence (vi+ 2, . . , ri _ J meets ri. 
(Here we use the simple connectivity of P and the maximahty of the rectangles). This 
would contradict that C is chordless. Therefore we can assume that ri_ lT\ri and 
Maximal rectangles q/a polyomino 213 
i+1 
i 
Fig. 1. Fig. 2. 
ri + I nri + 2 are crosses. The sides of ri crossed by ri _ 1 and the sides of ri + I crossed by 
ri+z cannot be parallel. Otherwise W, ,+ l,i would contradict the maximality of ri or 
ri+ I. Therefore these sides are perpendicular. (See Fig. 1.) We can shrink the rectangles 
ri and ri+ 1 so that their intersection become a cross. (See Fig. 2.) These rectangles are 
still maximal in the polyomino generated by the shrinked rectangles and the other 
rectangles of C. We have decreased the number of step intersections of C. In fact we 
can assume that all intersections are crosses. So, the cells ai can be chosen so that the a, 
are exactly the vertices of an orthogonal polygon. It is a well-known result that an 
orthogonal polygon admits a reflex vertex (its interior angle is equal to 3z/2) as soon 
as the number of its vertices is greater than 4. Let ai be this vertex. It is easy to check 
that ri cannot be maximal. Hence we have proved the following. 
Theorem 2. The length of a chordless cycle in G is at most 4. 
3. Star cutset or loose vertex 
Let L be a maximal rectangle of the polyomino P with the lowest top row. Let I be 
the top row of L. It is easy to see that the following holds. 
Proposition 2. Every rectangle which meets L meets also I. 
Now we shall show that if the neighbourhood of L in G contains a P, then 
G contains a star cutset. Let T,, T,, T3, T, be a P4 in the neighbourhood of L. 
Without loss of generality we assume that these rectangles are numbered so that the 
top row of T, is higher than the top row of T2. Then the low row of T2 is lower than 
the low row of T3. Otherwise T2 would meet T4. (See Fig. 3.) The set of vertices 
T3u(Tc(T3)\T4) is a star cutset deconnecting T1 and T4. Because if a chain of 
rectangles goes from T1 to T4 without meeting T3 then the top left corner of T3 would 
214 
Fig. 3 
be in the interior of P, contradicting the maximality of T3. This ends the proof of 
Theorem 1. 
References 
[l] J.B. Shearer, A class of perfect graphs, SIAM J. Alg. Discrete. Math. 3 (1982) 281-284. 
[2] F. Maire, Thesis, University Pierre et Marie Curie, Paris 6, 1993. 
[3] C. Berge, C.C. Chen, V. Chvatal and C.S. Seow, Combinatorial properties of polyominoes, 
Combinatorics 1 (1981) 217-224. 
[4] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1985). 
[S] C. Berge and V. ChvBtal, Topics on perfects graphs, Ann. Discrete Math. 21 (1984). 


file:///data/remote/core/dit/data/elsevier/pdf/642/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMDEyMzY1eDkzOTA1Nzho.pdf

A characterization of intersection graphs of the maximal rectangles of a polyomino

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector