MATHEMATICS

Theorem 2 Every graph G with n vertices is the intersection graph of a family of boxes in Euclidean n-space.

Proof: Let v1, v2, . . . , vn be the vertices of G. A box in Euclidean n-dimen- sional space is the set of all n-tuples of real numbers (x1, x2, . . . , xn) such that each xi is in some closed interval Ii. Now, for each k = 1 . . . , n and each vertex vi, define closed intervals Ik(vi) as follows.

Ik(vi) =

⎧⎪⎨ ⎪⎩

[0, 1] if i = k [1, 2] if i = k and {vi, vk} is an edge in G [2, 3] if i = k and {vi, vk} is not an edge in G.

For each vertex vi define a box B(vi) in Euclidean n-space by

B(vi) = {(x1, x2, . . . , xn) |xj ∈ Ij(vi) for j = 1, . . . , n}.

232 Applications of Discrete Mathematics

Thus, the box B(vi) corresponding to vi is the Cartesian product of the intervals Ij(vi) for j = 1, . . . , n.

Now we show that vi and vj are adjacent in G if and only if B(vi)∩B(vj) = ∅. Thus the graph G is the intersection graph of the family of boxes B(vi). First, suppose that there is an edge joining vl and vm. If k is different from both l and m, then according to the definition, Ik(vl)∩ Ik(vm) is [1, 2]∩ [1, 2], [1, 2]∩ [2, 3], or [2, 3] ∩ [2, 3]. In any case we have Ik(vl) ∩ Ik(vm) = ∅. If k=l or k=m then Ik(vl) ∩ Ik(vm) = [1, 2] ∩ [0, 1] = ∅. So, if there is an edge joining ve and vm, then for all k, Ik(vl) ∩ Ik(vm) = ∅. Hence B(vl) ∩ B(vm) = ∅.

Now suppose that B(vl)∩B(vm) = ∅. Then for each k from l to n, Ik(vl)∩ Ik(vm) = ∅. Set k = l then Il(vl) = [0, 1] and Il(vm) must be [1, 2] for the intersection to be nonempty. By definition of Il(vm), vl and vm are adjacent. Thus G is the intersection graph of the family of boxes B(vi).

This theorem shows that boxicity is well-defined. Unfortunately, there is no efficient algorithm known for determining the boxicity of a general graph. There is no characterization known for graphs of any specific boxicity other than 1.

In fact, there are not many general classes of graphs for which the boxicity is known. It is not hard to see that the boxicity of the n-cycle Cn is 2 for n = 4 or larger, and this is left as Exercise 6. Another general class of graphs for which the boxicity is known is the complete p-partite graphs. These are the graphs Kn1,n2,…,np defined as follows: there are n1 + · · ·+ np vertices partitioned into p classes, where the ith class has ni vertices. Within a class no vertices are adjacent, and every vertex in any class is adjacent to all vertices in the other classes. Roberts [6] showed that the boxicity of Kn1,…,np is equal to the number of ni that are larger than 1.

One result which helps somewhat in calculating the boxicity of a graph is due to Gabai [2]. This theorem depends on the concept of independence of a set of edges.

Definition 9 A set of edges in a graph is independent if they have no vertices in common.

Gabai’s theorem [2] is the following, stated without proof.

Theorem3 Let G be a simple graph. If the maximum size of an independent set of edges of G is k, then G has boxicity less than or equal to k. Also, if G has a generated subgraph consisting of k independent edges then the boxicity of G is greater than or equal to k.

Chapter 13 Food Webs 233

Gabai’s theorem is useful in determining the boxicity of relatively small graphs and for certain families. In any case it limits the amount of trial and error needed.

In our study of competition we search for the representation of the com- petition graph of a food web as the intersection graph of a family of sets in Euclidean n-space for some n. As a consequence of the theorem proved above, this representation is always possible. Furthermore, we can use the boxicity of the competition graph as an indicator of the minimum number of factors essential for describing competition in the community. Cohen [1] has studied more than 30 single-habitat food webs published in the ecological literature and has found that the competition graphs of all of them are interval graphs. That is, in all cases one dimension suffices to represent competition by niche overlap. It is not known whether this is a general law of ecology, but it does raise many interesting questions. In some single-habitat communities a single dimension for the niche space can be identified. It may be some obviously linear factor such as temperature, body length or depth in water. However, it may well be that more than one single dimension will work. And, of course, we can’t expect the single-niche dimension to be the same from community to community.

Hypothetical food webs have been constructed such that their competition graphs are not interval graphs, but these combinations of species have never been observed in nature at the same time and place.

The representation of graphs as intersection graphs of boxes has important applications in ecology, as we have seen. Applications to such diverse fields as archaeology and automobile traffic control have also been investigated (see reference [6]). We conclude with an additional application of food webs.

Trophic Status In the study of social systems it is often useful to measure the status of an individual in an organization. Harary [4] first introduced the idea of measuring the status of a species in a food web. In ecology this status is usually called the trophic level and is helpful in assessing the complexity and diversity of a web. The idea is that a web with many species at each trophic level has a high degree of complexity. In ecology it is generally thought that more complex ecosystems are more stable. In this section we study the question of how trophic status can be defined in a food web.

If the food web is simply a directed path (a food chain) then it is easy to define trophic status; just follow the order of the species in the chain. Some other structures also allow for an easy definition of trophic status. For example, we might think of species with no outgoing edges as being at the bottom of the web. Suppose that for every vertex, all directed paths to vertices at the bottom have the same length. Examples of such webs are given in Figure 8.

234 Applications of Discrete Mathematics

Figure 8. Graphs of two food webs.

In this kind of web, the trophic status of a vertex can be defined as the length of a directed path from the vertex to the bottom.

In general it is difficult to define trophic status in complicated food webs. Because more than one approach may be possible, we will use the term trophic status in this context rather than the term trophic level which is well-known in the context of food chains. Our goal is to investigate how trophic status could be measured rather than to develop a unique possibility.

To start, we need some basic assumptions about food webs. In particular, we assume that our food web is acyclic, i.e. that the directed graph has no cycles. Thus, there are no species s1, . . . , sn such that for i = 1, . . . , n − 1, si preys on si+1 and sn preys on s1. In particular there are no two species such that each preys on the other. Thus, the prey relationship is asymmetric.

We will take an axiomatic approach to defining measures of trophic status. That is, we will state conditions which any reasonable measure should satisfy in the form of axioms. A measure will then be acceptable if and only if it satisfies the axioms. The axioms will define an ideal model for the concept of measure of trophic status. Our approach will follow that of Harary [4] and Kemeny and Snell [5], who work with status in an organization, and the treatment in Roberts [6], which is more detailed.

Definition 10 In a food web a species v is a direct prey of a species u if there is a directed edge from u to v. A species v is an indirect prey of u if there is a directed path from u to v.

It could well happen that there are two species u and v neither of which is an indirect prey of the other.

Definition 11 If v is a direct or indirect prey of u, then the level of v relative to u is the length of the shortest directed path from u to v.

Chapter 13 Food Webs 235

We can now state some reasonable axioms for measures of trophic status. Let tW (u) be the measure of status in the food web W . The axioms are:

Axiom 1: If a species u has no prey then tW (u) = 0. Axiom 2: If, without otherwise changing the food web, we add a new vertex which is a direct prey of u to get a new web W ′, then tW ′(u) > tW (u). Axiom 3: Suppose the web W is changed by adding edges and/or vertices in such a way that the level of some direct or indirect prey of u is increased, and no direct or indirect prey of u has its level relative to u decreased. If W ′ is the new web, then tW ′(u) > tW (u).

These axioms make sense intuitively when we consider that we are saying that a species with no prey is at the bottom level (Axiom 1), that if the number of prey of a species increases its status increases (Axiom 2), and that the status of a species increases if its level relative to some indirect prey is increased (Axiom 3).

There is a measure of status which satisfies the axioms. Harary [4] sug- gested the following definition.

Definition 12 If a species u has nk species at level k relative to u for each k, then

hW (u) = ∑

k

knk.

Theorem 4 The measure hW (u) satisfies Axioms 1-3.

Proof: If u has no prey, then hW (u) = 0 because all the nk = 0. If we add a direct prey for u, then n1 increases by 1, so the sum defining

hW (u) also increases. Likewise, if some direct or indirect prey of u at level k relative to u is

moved to level k + n below u and no other direct or indirect prey of u has its level decreased, the sum for h increases by at least kn, verifying Axiom 3.

Kemeny and Snell [5] also show that if tW is any other measure of trophic status satisfying Axioms 1–3 and having all its values nonnegative, then for all species u, tW (u) ≥ hW (u). Thus, h is in a sense a minimal measure of trophic status.

While h satisfies our axioms it fails to have other desirable properties. For example, it seems reasonable that if tW is a measure of trophic status and v is a direct or indirect prey of u, then

tW (u) ≥ tW (v).

236 Applications of Discrete Mathematics

The measure h does not have this property. Figure 9 shows an example of an acyclic food web W with two vertices u and v for which v is a direct prey of u but hW (v) > hW (u).

Figure 9. An acyclic food web.

In this example, hW (u) = 6 and hW (v) = 8.

The problem we have found can be avoided if we modify our definition of level of one species relative to another: If v is a direct or indirect prey of u, then the level of v relative to u is the length of the longest directed path from u to v.

It is not hard to show that if h is defined by the same formula as before, but using the new definition of level, then h satisfies Axioms 1–3 as well as having the property that any species has higher status than any of its direct or indirect prey (see reference [5]). The problem we encountered here demonstrates one of the difficulties with the axiomatic approach. Our problem lay in the definition of level and this would not show up in any consideration of the reasonableness of the axioms. Ideally, all of the terms used in specifying the axioms should either be left undefined or else be checked for “reasonableness”, just as the axioms themselves are. In this light we would also have to examine the new definition of level.

Without referring to the notion of relative level in a food web, perhaps the only requirement we can state for a measure of trophic status is that if there is a directed path from u to v, then tW (u) ≥ tW (v).

There are other ways to investigate complexity of food webs and relative importance of species in food webs. General methods of measuring complexity in graphs can be applied to competition graphs and food webs. For example, such ideas as the number of edges divided by the number of vertices, and the average out-degree and average in-degree might be useful. The importance, or criticality, of a species in a food web could be studied by investigating what happens to the web when the species is deleted from the web. For example, if the web is disconnected when a species is removed that would indicated a high level of importance. More information on these questions can be found in [6].

Chapter 13 Food Webs 237

Suggested Readings

1. J. Cohen, Food Webs and Niche Space, Princeton University Press, Prince- ton, N.J., 1978.

2. H. Gabai, “N -dimensional Interval Graphs”, mimeographed, York College, C.U.N.Y., New York, 1974.

3. P. Gilmore and A. Hoffman, “A Characterization of Comparability Graphs and Interval Graphs”, Canadian J. Math., Vol. 16, 1964, pp. 539–548.

4. F. Harary, “Status and Contrastatus”, Sociometry , Vol. 22, 1959, pp. 23– 43.

5. J. Kemeny and J. Snell, Mathematical Models in the Social Sciences, MIT Press, Cambridge, MA, 1972.

6. F. Roberts, Discrete Mathematical Models with Applications to Social, Bi- ological, and Environmental Problems, Prentice Hall, Upper Saddle River, N.J., 1976.

Exercises

1. Find the ecological niche in Euclidean space of the appropriate dimensions in each case.

a) Temperature between 10oF and 90oF ; nitrate concentration in soil between 1% and 5%.

b) Carbon monoxide in atmosphere between 0% and 1%; relative humid- ity between 20% and 100%; nitrogen gas content in atmosphere between 15% and 20%.

2. Find the competition graph for the given food webs in each case: a) b)

238 Applications of Discrete Mathematics

3. Find a representation for each graph as the intersection graph of a family of rectangles in the plane.

a) b)

4. Find a representation for each graph as the intersection graph of a family of intervals on the line.

a) b)

5. Show that if a graph G is an interval graph then it satisfies the conditions of the theorem of Gilmore and Hoffman characterizing interval graphs. Hint: For an interval representation let I(v) be the interval assigned to the ver- tex v. If u and v are adjacent in G, make the orientation (u, v) if and only if I(u) lies entirely to the left of I(v).

6. Show that if Cn is the cycle of length n, then the boxicity of Cn is 1 for n = 3 and 2 for n ≥ 4.

7. According to Roberts’ result quoted in the text, the boxicity of the com- plete bipartite graph K(3, 3) is 2. Find a representation of K(3, 3) as the intersection graph of a family of boxes in the plane.

8. Let Q3 be the graph formed by the edges and corners of a cube in Euclidean three space. Is Q3 an interval graph? Why? Determine the boxicity of Q3.

9. A food web for some species in the Strait of Georgia, B.C. ([1], page 165) is given by the following table. The numbers atop columns indicate predator (consuming) species and those at the left of rows indicate prey (consumed) species. An entry 1 indicates that the predator in that column consumes the prey for that row, an entry 0 that it does not. The key identifies the various species.

Chapter 13 Food Webs 239

1 2 3 4 5 Key

2 1 0 0 0 0 1. Juvenile pink salmon 3 1 0 0 0 0 2. P. minutus 4 0 0 1 0 0 3. Calanus and Euphausiid furcilia 5 1 0 0 1 0 4. Euphausiid eggs 6 0 1 1 0 1 5. Euphausiids 7 0 0 1 0 1 6. Chaetoceros socialis and debilis

7. mu-flagellates

a) Construct a directed graph for this food web.

b) Construct the competition graph for this food web.

c) Find a set of intervals on the real line such that the graph of part b) is the intersection graph of this family of intervals.

10. Repeat Exercise 9 for the following food web for a community of pine feeders [1], p.148.

2 3 4 5 6 7 8 9 10 Key

1 1 1 0 0 0 0 0 0 0 1. Pine 2 0 0 1 1 0 0 0 0 0 2. Caterpillars, moths 3 0 0 1 0 1 1 1 1 1 3. Aphids, secretion 4 0 0 0 0 0 0 0 0 1 4. Digger wasps 5 0 0 0 0 0 0 0 0 1 5. Ichneumons 8 0 0 0 0 0 0 0 0 1 6. Bugs 9 0 0 0 0 0 0 0 0 1 7. Ants

10 0 0 1 0 0 0 0 0 0 8. Syrphids 9. Ladybugs

10. Spiders

11. Give an example of a food web which has two species, neither of which is a direct or indirect prey of the other.

12. In the section on trophic status two different definitions of relative level were given and two corresponding versions of the measure of trophic status hW were also given. Calculate the trophic status of each vertex in each of the following food webs using both versions of h.

240 Applications of Discrete Mathematics

a) b)

13. If the only requirement we make for a measure of trophic status tW is that if there is a directed path from u to v then tW (u) > tW (v), show that every acyclic food web has such a measure of trophic status.

14. (Roberts [6]) If relative level is measured using the length of the shortest directed path (our first definition), a plausible measure of trophic status is

tW (u) = ∑

v

hW (v),

where the sum is taken over all vertices v for which there is a directed path from u to v. Show that this possible measure has the property that if there is a directed path from u to v, then tW (u) ≥ tW (v). Which of the Axioms 1–3 does this measure satisfy?

15. In our discussion of trophic status we assumed that the food web was acyclic. How restrictive is this assumption? Can you think of two species each of which could have the other as prey?

Computer Projects

1. Write a program to calculate trophic status in acyclic food webs.

2. Write a program to calculate the adjacency matrix for the intersection graph of a family of intervals given as pairs (a, b) of their endpoints.

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