An independent set in a graph is a set of mutually non-adjacent vertices in the graph. So, no edge can have both its endpoints in an independent set. In this problem, we will count independent sets in ladder graphs. A ladder graph Ln is a graph obtained from the points and lines formed by a row of n squares. More formally: • its vertices correspond to ordered pairs (0,0), (0, 1), (1,0), (1, 1),..., (n, 0), (n, 1); • two vertices are adjacent if they represent points at distance exactly 1 from each other. The number n here is called the order of the ladder. Here is L3, a ladder of order 3: (0, 1) (1,1) (2,1) (3,1) (0,0) (1,0) Examples of independent sets in L3 include: Ø, {(2,1)}, {(0,0), (2,0), (3, 1)}, {(0, 1), (1,0), (2, 1), (3,0)). The simplest ladder is Lo, consisting of just the vertices (0,0) and (0,1) with a single edge between them. For all n, define (2,0) (3,0) an = number of independent sets in Ln which include neither (n, 0) nor (n,1); bn = number of independent sets in Ln which include (n, 0) but not (n,1); Cn = total number of independent sets in Ln. (c) Prove, by induction on n, that for all n ≥ 0, (a) Compute ao, bo, ai, b₁. (b) Give, with justification, recurrence relations that express an+1 and bn+1 in terms of an and bn. (Here, each of an+1 and bn+1 is expressed using both an and bn.) an < √2(√2+1)" and bn (√2+1)".
An independent set in a graph is a set of mutually non-adjacent vertices in the graph. So, no edge can have both its endpoints in an independent set. In this problem, we will count independent sets in ladder graphs. A ladder graph Ln is a graph obtained from the points and lines formed by a row of n squares. More formally: • its vertices correspond to ordered pairs (0,0), (0, 1), (1,0), (1, 1),..., (n, 0), (n, 1); • two vertices are adjacent if they represent points at distance exactly 1 from each other. The number n here is called the order of the ladder. Here is L3, a ladder of order 3: (0, 1) (1,1) (2,1) (3,1) (0,0) (1,0) Examples of independent sets in L3 include: Ø, {(2,1)}, {(0,0), (2,0), (3, 1)}, {(0, 1), (1,0), (2, 1), (3,0)). The simplest ladder is Lo, consisting of just the vertices (0,0) and (0,1) with a single edge between them. For all n, define (2,0) (3,0) an = number of independent sets in Ln which include neither (n, 0) nor (n,1); bn = number of independent sets in Ln which include (n, 0) but not (n,1); Cn = total number of independent sets in Ln. (c) Prove, by induction on n, that for all n ≥ 0, (a) Compute ao, bo, ai, b₁. (b) Give, with justification, recurrence relations that express an+1 and bn+1 in terms of an and bn. (Here, each of an+1 and bn+1 is expressed using both an and bn.) an < √2(√2+1)" and bn (√2+1)".
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Discrete math
Transcribed Image Text:An independent set in a graph is a set of mutually non-adjacent vertices in the graph. So, no edge
can have both its endpoints in an independent set.
In this problem, we will count independent sets in ladder graphs. A ladder graph Ln is a graph
obtained from the points and lines formed by a row of n squares. More formally:
• its vertices correspond to ordered pairs (0,0), (0, 1), (1,0), (1, 1),..., (n, 0), (n, 1);
• two vertices are adjacent if they represent points at distance exactly 1 from each other.
The number n here is called the order of the ladder. Here is L3, a ladder of order 3:
(0,1)
(1,1)
(2,1)
(3,1)
(1,0)
Examples of independent sets in L3 include: Ø, {(2,1)}, {(0,0), (2,0), (3, 1)}, {(0, 1), (1,0), (2, 1), (3,0)}.
The simplest ladder is Lo, consisting of just the vertices (0,0) and (0,1) with a single edge
between them.
For all n, define
(0,0)
(2,0)
an = number of independent sets in Ln which include neither (n, 0) nor (n, 1);
bn = number of independent sets in Ln which include (n, 0) but not (n,1);
Cn = total number of independent sets in Ln.
(c) Prove, by induction on n, that for all n ≥ 0,
(3,0)
(a) Compute ao, bo, a1, b₁.
(b) Give, with justification, recurrence relations that express an+1 and bn+1 in terms of an and bn.
(Here, each of an+1 and bn+1 is expressed using both an and bn.)
an < √2(√2+1)" and
bn ≤ (√2+1)".
(d) Hence give a good closed-form upper bound for en (i.e., an upper-bound expression in terms
of n, not a recurrence relation). Your bound should be significantly better than 3" (which is the
upper bound you get by ignoring horizontal edges and only taking vertical edges into account).
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