Explicit Combinatoric Structures of Palindromes and Chromatic Number of Restriction Graphs

TL;DR

The paper investigates reconstruction of strings from palindromic fingerprints $PF(S)$, the set of all maximal palindromic substrings, establishing tight combinatorial bounds and an explicit structure for the shortest fingerprints that force at least $k$ distinct characters. It introduces the restriction-graph framework, linking reconstruction to graph coloring, and proves a formal bound $IPF(k)=1$ if $k=1$ and $IPF(k)=2^{k-2}+1$ otherwise, solving an open problem. A key insight is that the reconstruction alphabet size is bounded by $ ext{chi}(G)\,ig( ext{the chromatic number of the restriction graph}ig) \, ext{and}\, ext{chi}(G)\,\le\, ext{log}(n-1)+2$, along with an explicit recursive construction $S_k$ of the shortest preimage for non-crossing fingerprints given by $S_k=xLtLy$ with $L$ palindrome and $x,t,y$ new symbols. The authors further extend to fingerprints with crossing palindromes by showing a merging argument that reduces general cases to non-crossing ones, establishing that the restriction graph of minimal fingerprints is a clique and that the reconstruction problem is equivalent to a graph-coloring instance. This work provides actionable combinatorial structure for palindromic fingerprint reconstruction and connects it to chromatic-graph theory, with potential implications for data compression and alphabet-size control in pattern-based reconstructions.

Abstract

The palindromic fingerprint of a string $S[1\ldots n]$ is the set $PF(S) = \{(i,j)~|~ S[i\ldots j] \textit{ is a maximal }\\ \textit{palindrome substring of } S\}$. In this work, we consider the problem of string reconstruction from a palindromic fingerprint. That is, given an input set of pairs $PF \subseteq [1\ldots n] \times [1\ldots n]$ for an integer $n$, we wish to determine if $PF$ is a valid palindromic fingerprint for a string $S$, and if it is, output a string $S$ such that $PF= PF(S)$. I et al. [SPIRE2010] showed a linear reconstruction algorithm from a palindromic fingerprint that outputs the lexicographically smallest string over a minimum alphabet. They also presented an upper bound of $\mathcal{O}(\log(n))$ for the maximal number of characters in the minimal alphabet. In this paper, we show tight combinatorial bounds for the palindromic fingerprint reconstruction problem. We present the string $S_k$, which is the shortest string whose fingerprint $PF(S_k)$ cannot be reconstructed using less than $k$ characters. The results additionally solve an open problem presented by I et al.

Figures (4)

• Figure 1: The palindromes in $S$
• Figure 2: The equality graph of $S$.
• Figure 3: $v_1=\{9\},v_2=\{12,14\},v_3=\{13\},v_4=\{5\},v_5=\{1\}, v_6=\{10,11\},v_7=\{2,4,6,8\},v_8=\{3,7\}$
• Figure 4: The palindromes and islands of $S$

Theorems & Definitions (62)

• Definition 1: Palindromic Fingerprint
• Theorem 2
• Definition 3
• Definition 4: Trivial palindrome
• Definition 5: Palindromic descriptor
• Definition 6
• Definition 7
• Example 8
• Theorem 8
• Example 9