1.2 Distance in Graphs
1.2.1 Definitions and a Few Properties
I prefer the term ‘eccentric.’
— Brenda Bates, Urban Legend Distance in graphs is defined in a natural way: in a connected graphG, the dis- tance from vertexuto vertexvis the length (number of edges) of a shortestu–v path inG. We denote this distance byd(u, v), and in situations where clarity of context is important, we may writedG(u, v). In Figure 1.26,d(b, k) = 4and d(c, m) = 6.
a
j m
l n i
k h
g f e
c d
b
FIGURE 1.26.
For a given vertexvof a connected graph, the eccentricity ofv, denotedecc(v), is defined to be the greatest distance fromvto any other vertex. That is,
ecc(v) = max
x∈V(G){d(v, x)}.
In Figure 1.26,ecc(a) = 5since the farthest vertices froma(namelyk,m,n) are at a distance of 5 froma.
Of the vertices in this graph, verticesc,k,mandnhave the greatest eccentricity (6), and verticese,f andghave the smallest eccentricity (3). These values and types of vertices are given special names. In a connected graphG, the radius ofG, denotedrad(G), is the value of the smallest eccentricity. Similarly, the diameter ofG, denoteddiam(G), is the value of the greatest eccentricity. The center of the graphGis the set of vertices,v, such thatecc(v) = rad(G). The periphery ofG is the set of vertices,u, such thatecc(u) = diam(G). In Figure 1.26, the radius is 3, the diameter is 6, and the center and periphery of the graph are, respectively, {e, f, g}and{c, k, m, n}.
Surely these terms sound familiar to you. On a disk, the farthest one can travel from one point to another is the disk’s diameter. Points on the rim of a disk are on the periphery. The distance from the center of the disk to any other point on the disk is at most the radius. The terms for graphs have similar meanings.
Do not be misled by this similarity, however. You may have noticed that the diameter of our graphGis twice the radius ofG. While this does seem to be a natural relationship, such is not the case for all graphs. Take a quick look at a cycle or a complete graph. For either of these graphs, the radius and diameter are equal!
The following theorem describes the proper relationship between the radii and diameters of graphs. While not as natural, tight, or “circle-like” as you might hope, this relationship does have the advantage of being correct.
Theorem 1.4. For any connected graphG,rad(G)≤diam(G)≤2 rad(G).
Proof. By definition,rad(G) ≤ diam(G), so we just need to prove the second inequality. Letuandvbe vertices inGsuch thatd(u, v) = diam(G). Further, let cbe a vertex in the center ofG. Then,
diam(G) =d(u, v)≤d(u, c) +d(c, v)≤2 ecc(c) = 2 rad(G).
The definitions in this section can also be extended to graphs that are not con- nected. In the context of a single connected component of a disconnected graph, these terms have their normal meanings. If two vertices are in different compo- nents, however, we say that the distance between them is infinity.
We conclude this section with two interesting results. Choose your favorite graph. It can be large or small, dense with edges or sparse. Choose anything you like, as long as it is your favorite. Now, wouldn’t it be neat if there existed a graph in which your favorite graph was the “center” of attention? The next theorem (credited to Hedetneimi in [44]) makes your wish come true.
Theorem 1.5. Every graph is (isomorphic to) the center of some graph.
Proof. LetGbe a graph (your favorite!). We now construct a new graphH (see Figure 1.27) by adding four vertices (w,x,y,z) toG, along with the following edges:
{wx, yz} ∪ {xa|a∈V(G)} ∪ {yb|b∈V(G)}.
Now,ecc(w) = ecc(z) = 4,ecc(y) = ecc(x) = 3, and for any vertexv∈V(G),
w x y z
G
FIGURE 1.27.Gis the center.
ecc(v) = 2. Therefore,Gis the center ofH.
Suppose you don’t like being the center of attention. Maybe you would rather your favorite graph avoid the spotlight and stay on the periphery. The next theorem (due to Bielak and Sysło, [25]) tells us when that can happen.
Theorem 1.6. A graphGis (isomorphic to) the periphery of some graph if and only if either every vertex has eccentricity1, or no vertex has eccentricity1.
Proof. Suppose that every vertex ofGhas eccentricity 1. Not only does this mean thatGis complete, it also means that every vertex ofGis in the periphery.Gis the periphery of itself!
On the other hand, suppose that no vertex ofGhas eccentricity 1. This means that for every vertexuofG, there is some vertexvofGsuch thatuv ∈ E(G).
Now, letH be a new graph, constructed by adding a single vertex,w, toG, to- gether with the edges{wx|x∈V(G)}. In the graphH, the eccentricity ofwis 1 (wis adjacent to everything). Further, for any vertexx∈V(G), the eccentricity ofxinHis 2 (no vertex ofGis adjacent to everything else inG, and everything inGis adjacent tow). Thus, the periphery ofHis precisely the vertices ofG.
For the reverse direction, let us suppose thatGhas some vertices of eccentricity 1 and some vertices of eccentricity greater than 1. Suppose also (in anticipation of a contradiction) thatGforms the periphery of some graph, sayH. Since the eccentricities of the vertices inGare not all the same, it must be thatV(G)is a proper subset ofV(H). This means thatH is not the periphery of itself and thatdiam(H) ≥2. Now, letv be a vertex ofGwhose eccentricity inGis 1 (v is therefore adjacent to all vertices ofG). Sincev ∈ V(G)and sinceGis the periphery ofH, there exists a vertexwinHsuch thatd(v, w) = diam(H)≥2.
The vertexw, then, is also a peripheral vertex (see Exercise 4) and therefore must be inG. This contradicts the fact thatvis adjacent to everything inG.
Exercises
1. Find the radius, diameter and center of the graph shown in Figure 1.28.
FIGURE 1.28.
2. Find the radius and diameter of each of the following graphs:P2k,P2k+1, C2k,C2k+1,Kn,Km,n.
3. For each graph in Exercise 2, find the number of vertices in the center.
4. Ifxis in the periphery ofGandd(x, y) = ecc(x), then prove thatyis in the periphery ofG.
5. Ifuandv are adjacent vertices in a graph, prove that their eccentricities differ by at most one.
6. A graphGis called self-centered ifC(G) =V(G). Prove that every com- plete bipartite graph, every cycle, and every complete graph is self-centered.
7. Given a connected graphGand a positive integerk, thekth power ofG, denotedGk, is the graph withV(Gk) =V(G)and where verticesuandv are adjacent inGk if and only ifdG(u, v)≤k.
(a) Draw the 2nd and 3rd powers ofP8andC10. (b) For a graphGof ordern, what isGdiam(G)?
8. (a) Find a graph of order 7 that has radius 3 and diameter 6.
(b) Find a graph of order 7 that has radius 3 and diameter 5.
(c) Find a graph of order 7 that has radius 3 and diameter 4.
(d) Supposeranddare positive integers andr ≤ d ≤ 2r. Describe a graph that has radiusrand diameterd.
9. Suppose thatuandvare vertices in a graphG,ecc(u) =m,ecc(v) =n, andm < n. Prove thatd(u, v) ≥ n−m. Then draw a graphG1where d(u, v) =n−m, and another graphG2whered(u, v)> n−m. In each case, label the verticesuandv, and give the values ofmandn.
10. LetGbe a connected graph with at least one cycle. Prove thatGhas at least one cycle whose length is less than or equal to2 diam(G) + 1.
11. (a) Prove that ifGis connected anddiam(G)≥3, thenGis connected.
(b) Prove that ifdiam(G)≥3, thendiam(G)≤3.
(c) Prove that ifGis regular anddiam(G) = 3, thendiam(G) = 2.