Minimal Unit Circular-Arc Models are Integer

(1)

Minimal Unit Circular-Arc Models are Integer

Francisco J. Soulignac & Pablo Terlisky CONICET, Universidad de Quilmes, Universidad de Buenos Aires {fsoulign,terlisky}@dc.uba.ar

Introduction

I A proper circular-arc (PCA) model M is defined based on the endpoints s(A) and t(A) of its arcs on a circle C.

0

t(A₁) s(A₁)

A₁

t(A₃)

s(A₃)

A₃

0 2 4

6 8 12 10

14 16 18

20

22 0 2

4 6 8 12 10

14 16 18

20 22

PCA model (24, 7)-CA model 7-IG model I (c, `)-CA model: C has circumference c, arcs have length `.

I `-IG model: arcs have length `, no arc crosses 0.

Problem: Find the Minimal Model

I Given a PCA model, find an equivalent

“minimal” model.

I A (c, `)-CA model M is weakly minimal when 1. `_M ≤ `_N and

2. c_M ≤ c_N .

For any equivalent model N . If also 3. s_M(A_i) ≤ s_N (A_i) for 1 ≤ i ≤ n,

then M is strongly minimal.

I Remark: the existence of minimal models is not obvious because 1–3 must hold simultaneously.

I Pirlot: every PIG model is equivalent to an integer strongly minimal model.

I Soulignac: every UCA model is equivalent to a weakly minimal model. Is it integer?

I Conjecture: c, ` ∈ N for every minimal (c, `)-CA model.

0 1 2 3

4 5 7 6

9 8 10 11 12

131415

A

B C

D

E F

G

0 1 2 3

4 5 7 6 9 8

1110 12 13

14151617

A

B

C

D E

F

G

Figure 1: a PCA model and its equivalent minimal model.

Synthetic Graphs (Pirlot, Soulignac)

I For a PCA model M with arcs A₁ < . . . < A_n: . One vertex for each arc A₁, . . . , A_n.

. steps: A_i → A_i+1 for 1 ≤ i ≤ n.

. noses: v(A_i) → v(A_j) when t(A_i)s(A_j) are consecutive in M.

. hollows: v(A_i) → v(A_j) when s(A_i)t(A_j) are consecutive in M.

. weights: separation of the endpoints in any equivalent (c, `)-CA model.

I Theorem: equivalent to (c, `)-CA iff no cycle has a positive weight.

Mitas’ plane drawing of the synthetic graph

I Vertices: positions in a circular matrix (first column follows last column).

I Internal edges (thick): advance to the right and induce a plane

drawing.

I External edges (thin): escape through the top and last rows.

I Consequently, every cycle has exactly one more hollow than nose.

I Hence, every cycle W has weight (` + 1) + |W|

. implying that the minimum feasible ` is integer.

I We exploit this drawing to settle Soulignac’s conjecture.

A new characterization of UCA models

I Intervalization I(M, k): copy k times the arcs of M in a PIG model.

. Nose (hollow) path: contain no hollows (no noses).

. Nose-(hollow-)like: more ext. noses (hollows) than ext. hollows (noses).

. Greedy nose (hollow): prefer noses (hollows) to steps.

I Characterization of UCA models.

. M is equivalent to a UCA model.

. every nose-like circuit intersects every hollow-like circuit.

. any greedy nose cycle intersects any greedy hollow cycle.

I Proof. Take advantage of the plane drawing of S(I(M, k)).

Figure 2: Plane drawing of S(I(M, 3)) for the model M in Figure 1.

I Simpler O(n) time recognition algorithm of UCA graphs.

Minimal UCA models are integer

I Let S be the synthetic graph of a minimal (c, `)-CA model M.

I Fact. S has nose-like W_N and hollow-like W_H cycles with weight 0.

I From the common vertex of W_N and W_H, we build a circuit of S with weight 0 that corresponds to a cycle of S(I(M, k)).

. Hence, ` coincides with the minimal feasible length for I(M, k).

. Implying that ` is integer.

I Next, we build a nose-like circuit of S with one nose more than hollows.

. Implying that c is integer.

I The algorithm to find M runs in O(n³) time and O(n²) space.

Conclusions

I We define the intervalization I of a PCA model M.

. Simpler characterization of UCA models.

. Different properties about S(M) are found by studying S(I).

I We prove that ` and c are integer for minimal (c, `)-CA models.

I We devise an O(n³) time algorithm to compute a minimal (c, `)-CA model equivalent to an input UCA model.

References

[1] Jutta Mitas.

Minimal representation of semiorders with intervals of same length.

In Orders, algorithms, and applications (Lyon, 1994), volume 831 of Lecture Notes in Comput. Sci., pages 162–175. Springer, Berlin, 1994.

[2] Francisco J. Soulignac.

Bounded, minimal, and short representations of unit interval and unit circular-arc graphs.

CoRR, abs/1408.3443, 2014.

[3] Marc Pirlot.

Minimal representation of a semiorder.

Theory and Decision, 28(2):109–141, 1990.

Acknowledgments

I Supported by PICT ANPCyT grants 2013-2205 and 2015-2419, and by PUNQ Grant 1451/15.