Sistema de manejo de los residuos sólidos

In this subsection we survey and extend results on some special graph classes that will be of interest to us. We start with a few definitions:

Definition 3.3.1. A graph is a comparability graphif its edges admit a transitive orientation. In other words, one can direct the edges in such a way that for any directed path xyz, there is a directed edge (arc) xz. A graph is a co-comparability graphif it is the complement of a comparability graph.

We will define permutation graphs in more detail later. For now, we will use the following characterisation:

Theorem 3.3.1 ([Dushnik and Miller, 1941]). A graph is a permutation graph

if it is comparability and co-comparability.

Definition 3.3.2. For a graph G and vertices v, w∈ G, we say that v dominates w if N(w)\ {v} ⊆N(v). It is easy to check that domination defines a quasi-order onV(G). This quasi-order will be called the vicinal order.

Definition 3.3.3. The Dilworth number dilw(G) of a graph G is defined as the minimum number of chains in any partition ofV(G) into chains with respect to the vicinal order. The same definition can be adapted to subsets of V(G).

Remark. By Theorem 1.2.1 [Dilworth, 1950], dilw(G) is the maximum size of an antichain inV(G) with respect to the vicinal order.

Example 3.3.1. A bipartite graph such that each part has Dilworth number 1 is called a bipartite chain graph. It is trivial to show that the class of bipartite chain graphs is precisely the class of2K2-free bipartite graphs.

Figure 3.2: A bipartite chain graph.

Definition 3.3.4. Asplit graphis a graphGwhose vertex setV(G)is partitionable into a clique and an independent set.

The class of split graphs is clearly hereditary and has a well-known characteri- sation in terms of forbidden induced subgraph, due to F¨oldes and Hammer:

Theorem 3.3.2 ([F¨oldes and Hammer, 1977a]). The class of split graphs is given by F ree(2K2, C4, C5). In other words a graph is a split graph iff it is chordal

and co-chordal.

There is an obvious analogy between split graphs and bipartite graphs. In order to relate properties of these two classes, we will define the following notation to be used in this section:

Definition 3.3.5. For a split graphG, denote by β(G) the bipartite graph obtained fromG by replacing its clique-part with an independent set.

Remark. If we think of β as a function from the class of split graphs to the class of bipartite graphs, then β is clearly surjective. Also note that for an arbitrary bipartite graphH, the setβ−1(H) will be a well-defined subset of split graphs with cardinality 1 or 2. For a set of split graphsGwe will denoteβ(G) :={β(G) :G∈ G}. For a set of bipartite graphsH, we will denote β−1(H) :=∪_H∈H β−1(H).

Definition 3.3.6. A graph G is a threshold graph if there exists a real number S and a real weightw(v) for each v∈V(G) such that

E(G) ={((u, v)∈V(G)2 _{:u6=v andw(u) +w(v)≥S}.}

Threshold graphs are motivated by applications in several disciplines such as psychology, computer science and scheduling [McKee and McMorris, 1999]. The following characterisation is due to Chv´atal and Hammer:

Theorem 3.3.3 ([Chv´atal and Hammer, 1973]). A graph is a threshold graph iff it is a split cograph, i.e. a graph belonging to F ree(2K2, P4, C4, C5).

Claim 3.3.4. The class G of threshold graphs corresponds to bipartite chain graphs in the sense thatH:=β(G)is the class of2K2-free bipartite graphs andβ−1(H) =G.

Proof. This follows trivially from the observation that a split graphGis P4-free iff

β(G) is 2K2-free.

Given the claim, the following theorem will not come as a surprise to the reader. Threshold graphs are precisely those graphs that form a chain with respect to the vicinal order:

Theorem 3.3.5 ([F¨oldes and Hammer, 1978]). A graphGis a threshold graph iff dilw(G) = 1.

A natural question is to ask for a characterisation of graphs G with Dilworth number at most two.

Definition 3.3.7. A graph is a threshold signed graph (TS-graph) if there exist real numbers S, T and a real weightw(v) for each v∈V(G) such that

E(G) ={((u, v)∈V(G)2 :u6=v and(w(u) +w(v)≥S or |w(u)−w(v)| ≤T)}

Benzaken et a.l. proved that this slight generalisation of threshold graphs gives the class of graphs with Dilworth number at most two:

Theorem 3.3.6 ([Benzaken et al., 1985a]). G is a TS-graph iff dilw(G)≤2.

Recall that class of split graphs is the intersection of chordal graphs and co- chordal graphs. Also, the class of permutation graphs is the intersection of compa- rability graphs and co-comparability graphs [Dushnik and Miller, 1941]. We quote one more result:

Proposition 3.3.7 ([Gilmore and Hoffman, 1964]). Interval graphs are exactly the chordal co-comparability graphs.

There are many equivalent ways to characterise split TS-graphs:

Theorem 3.3.8. The following are equivalent: 1. Gis a split graph with dilw(G)≤2. 2. Gis a split TS-graph.

3. Gis a split graph that is both an interval graph and a comparability graph. 4. Gis a split permutation graph.

5. Gis both an interval graph and a co-interval graph.

6. Gis a (3-sun, co-3-sun, rising sun, co-rising sun)-free split graph.

The equivalence of 1. and 2. is just a restatement of Theorem 3.3.6. The equivalence of 4. and 5. follows directly from Proposition 3.3.7. Since the same proposition implies that a split graph is an interval graph iff it is a co-comparability graph, the equivalence of 3. and 4. is also immediate. The proof of the equivalence of 1. and 3. is given in [F¨oldes and Hammer, 1977b]. Finally, the proof of the equivalence of 1. and 6. is from [Akiyama et al., 1983].

In a much-quoted paper [Benzaken et al., 1985b], the authors claim to give a shorter proof to the result in [Akiyama et al., 1983], having found a forbidden in- duced subgraph characterisation for the class of TS-graphs. For the reader’s benefit, we note that in the same paper (Theorem 5), the authors misquote a result from the paper by F¨oldes and Hammer, making an error in justifying the equivalence of 2. and 5.

From this point forward, we will generally refer to split TS-graphs as split per- mutation graphs.

Definition 3.3.8. • A bichain graph is a bipartite graph such that at least one part can be partitioned into at most two chains with respect to the vicinal order. • A double bichain graph is a bipartite graph such that each part can be parti-

tioned into at most two chains with respect to the vicinal order.

We claim that there is a natural correspondence between split permutation graphs and double bichain graphs:

Proposition 3.3.9. Let G denote the class of split permutation graphs. Then β(G)

is the class of double bichain graphs and β−1(β(G)) is the class of split permuta- tion graphs. Furthermore, double bichain graphs are precisely the (3K2, C6, P7)-free

bipartite graphs.

Proof. Let G be a split permutation graph. By Theorems 3.3.5 and 3.3.8, G can be partitioned into two threshold graphs G1 and G2, each of which is a chain in G with respect to the vicinal order. But then β(G1) and β(G2) will offer a suitable partition of β(G), proving that it is a double bichain graph. Conversely, for any double bichain graph H, one can easily partition any graph in β−1(H) into two threshold graphs, each forming a chain with respect to the vicinal order.

To prove the forbidden induced subgraph characterisation for double bichain graphs, it suffices to refer to Theorem 3.3.8, noting that

β(3-sun, co-3-sun, rising sun, co-rising sun) = (3K2, C6, P7) and

β−1(3K2, C6, P7) = (3-sun, co-3-sun, rising sun, co-rising sun). Clearly a split graph

Gis (3-sun, co-3-sun, rising sun, co-rising sun)-free iff β(G) is (3K2, C6, P7)-free. For completeness, let us give a direct proof for the forbidden induced subgraph characterisation of double bichain graphs. This will also allow us to make some observations about the respective characterisation for bichain graphs.

Proposition 3.3.10. The class of double bichain graphs is the class of(3K2, C6, P7)-

free bipartite graphs.

Proof. To prove the theorem, we will show that a bipartite graph G = (U, V, E) with a bipartitionU∪V is (P7, C6,3K2)-free if and only if the vertices of each part of the graph can be partitioned into at most 2 vicinal chains.

One direction of the proof is simple, because at least one part in each of the graphsP7, C6 and 3K2 contains three vertices which are incomparable with respect to the vicinal order.

Now assume thatGis (P7, C6,3K2)-free. Suppose, for contradiction, that in one part of G, say U, there is an antichain of three vertices a, b, c with respect to the vicinal order. Then, in the part V, there exists a vertex d which is adjacent to a

but notb, and a vertex e which is adjacent to b but nota. We will split the proof into three cases:

Case 1. Supposec is adjacent to both dand e. Then there must exist a vertex

f which is adjacent to a but notc. Vertex b must be non-adjacent tof, otherwise

tobbut non-adjacent toc. Again, to avoid an inducedC6, the vertexgmust also be non-adjacent toa. But thenf adcebg forms an inducedP7, which is a contradiction.

Case 2. Now suppose that c is adjacent to exactly one of dand e, saye. Then there must exist a vertexg which is adjacent tobbut notc, and a vertexhwhich is adjacent tocbut notb. Ifawere adjacent to neither ofgand h, thenadbgchwould form an induced 3K2. If a were adjacent to exactly one of g and h, say g, then

dagbechwould form an induced P7. Finally, if ais adjacent to both g and h, then

agbechwould form an inducedC6. Each of these possibilities is a contradiction.

Case 3. Finally, suppose c is non-adjacent to d and e. Then there must exist a vertex h which is adjacent to c but not b, and a vertex i which is adjacent to c

but nota. Note that h and i must not be the same vertex, since otherwiseadbech

would form an induced 3K2. The vertexamust be adjacent toh, otherwiseadbech would form an induced 3K2. Similarly, bmust be adjacent toi. Nowdahcibeforms an inducedP7, which is a contradiction.

We have exhausted all possible cases, each leading to a contradiction. Thus our proof is complete.

Note that all of the contradicting copies of P7 found in the proof of Proposi- tion 3.3.10 have the same bi-coloring with respect to the considered (P7, C6,3K2)- free bipartite graphG. Thus we obtain the following immediate corollary:

Corollary 3.3.11. The class of (bichain) graphs G := (U, V, E) such that U has Dilworth number at most 2 is precisely the class of (P₇U, C6,3K2)-free bipartite

graphs, wherePU

7 is a copy of P7 containing exactly three vertices inU.

Trivially, the class of double bichain graphsG:= (U, V, E) is merely the intersec- tion of the two classes of (P₇U, C6,3K2)-free bipartite graphs and (P7V, C6,3K2)-free bipartite graphs. We can make an analogous statement for the class of bichain graphs:

Corollary 3.3.12. The class of bichain graphs G:= (U, V, E) is the union of the two classes of (PU

7 , C6,3K2)-free bipartite graphs and (P7V, C6,3K2)-free bipartite

graphs, where P₇U and P₇V are defined as in the the previous corollary. Thus the class of bichain graphs is the class of {3K2, C6} ∪ S-free bipartite graphs, where S

is the set of minimal graphs containing copies of bothPU

7 and P7V.

Determining the set S in Corollary 3.3.12 would result in a forbidden induced subgraph characterisation of bichain graphs.

• β(co-rising sun) =P₇U • β(rising sun) =P₇V • β(3-sun) =C6

• β(co-3-sun) = 3K2

In [F¨oldes and Hammer, 1977b], it is shown that split interval graphs (i.e. split co-comparability graphs) are the (3-sun, co-3-sun, rising sun)-free split graphs. Analogously, the split co-interval graphs (i.e. split comparability graphs) are the (3-sun, co-3-sun, co-rising sun)-free split graphs.

In other words, split interval graphs are the split graphs whose ’independent set’-part has Dilworth number at most 2. Analogously, the split co-interval graphs are the split graphs whose clique-part has Dilworth number at most 2.

We may thus deduce that determining the setS in Corollary 3.3.12 would also result in a forbidden induced subgraph characterisation of the class of all split graphs that are either comparability or co-comparability.