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The Road Problem - Special Case Study

The Road Problem - Special Case Study. EE392P Final Project Dec 2000 HaiXin Tie. Outline. The road problem statement Known results Some special case study Conjecture Future directions. The Road Problem. Let G be an unlabeled graph, assume: G is primitive

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The Road Problem - Special Case Study

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  1. The Road Problem- Special Case Study EE392P Final Project Dec 2000 HaiXin Tie

  2. Outline • The road problem statement • Known results • Some special case study • Conjecture • Future directions

  3. The Road Problem • Let G be an unlabeled graph, assume: • G is primitive • Every state in G has an out-degree 2 • Then: • There is a deterministic binary labeling (or coloring) for G • There exist a homing word for that labeled graph

  4. Known Results • “On the Road Coloring Problem” by Joel Friedman, 90 • A: Adjacency Matrix of G • A  1 = 2 1 • 2: eigenvalue, 1: right eigenvector • Exist a left eigenvector for 2: w, w A = 2 w • Define: • w(v) = the weight of state v • W = w(1) + w(2) + … + w(n) = the weight of the graph • THEOREM: For any (+/-) labeling, exist i and a partition of states T0, T1, T2, … Ti-1, each of which is collapsible, and has weight w0. In particular, w0 i = W • THEOREM: Let G have a coloring with a + tree and a + cycle of length m. Then i, the size of a minimal image, divides m. • COROLLARY: If (m, W) = 1, then G is collapsible • COROLLARY: If W is a prime power, then G is collapsible

  5. Special Case: In-Splitting • Lemma 1: Any primitive graph with out-degree 2 that is generated through a series of in-splitting has a homing word. • Proof: Induction.

  6. Small Graphsn = 1, 2 (with a self-loop) In-Splitting Generalized In-Splitting

  7. n = 3 (without self-loop) Generalized In-Splitting

  8. n = 3 (another way of splitting) Generalized In-Splitting

  9. n = 3 (cont.) Generalized In-Splitting(s) ?

  10. n = 4, 5, 6…(?) • Complexity grows exponentially • Possible ways for states-splitting • Primitivity check of the graph

  11. Special Case+: G-In-Splitting • Lemma 2: Any primitive graph with out-degree 2 that is generated through a series of primitive generalized-in-splitting has a homing word.

  12. Observations • Exist other special cases that does not seem apparent from G-In-Splitting (if they really are); • “Coloring with red tree” can not be extended to the general case; • Homing words becomes more rare as n increases, but it also largely depends on how to in-splitting is conducted; • Many graphs (all?) can be generated from graphs with smaller number of states with reservation of at least one homing word.

  13. Conjecture • All the primitive binary graph can be generated from smaller graphs, and its homing word can be extended from one of the homing words in that smaller graph. (This could be a constructive way to find a homing word)

  14. Future direction • To extent the state-splitting algorithm to n = 4, 5, … • To find a way to categorize all types of graphs, and to generalize the induction to any n • If the conjecture is true, it can be a good method for cryptography.

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