Succinct Data Structure for Graphs with $d$-Dimensional $t$-Representation

TL;DR

The paper tackles succinct data structures for graphs with a d-dimensional t-representation. By combining a counting-based lower bound with a carefully constructed subclass, it proves a near-tight lower bound on the number of such graphs and presents a succinct data structure using ((2dt-1)n log n + 2dtn log t + o(dtn log n)) bits that supports adj and neighbor queries in polylogarithmic time. The results are succinct when td^2 = o(n/log n) and yield immediate corollaries for graphs of bounded boxicity and bounded interval number, recovering and extending known interval-graph results. Additionally, a conditional hardness result based on the Boolean matrix multiplication conjecture establishes limits on neighbor-query speed for large-t interval representations. Overall, the work advances succinct representations beyond interval graphs and provides practical schemes for t-interval and d-boxicity graphs under a scalable sparsity regime.

Abstract

Erdős and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting of at most $t$ $d$-dimensional axis-parallel boxes. In particular, for a graph $G$ and for each $d \geq 1$, they consider $i_d(G)$ to be the minimum $t$ for which $G$ has such a representation. For fixed $t$ and $d$, they consider the class of $n$ vertex labeled graphs for which $i_d(G) \leq t$, and prove an upper bound of $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4πt)$ on the logarithm of size of the class. In this work, for fixed $t$ and $d$ we consider the class of $n$ vertex unlabeled graphs which have a {\em $d$-dimensional $t$-representation}, denoted by $\mathcal{G}_{t,d}$. We address the problem of designing a succinct data structure for the class $\mathcal{G}_{t,d}$ in an attempt to generalize the relatively recent results on succinct data structures for interval graphs (Algorithmica'21). To this end, for each $n$ such that $td^2$ is in $o(n / \log n)$, we first prove a lower bound of $(2dt-1)n \log n - O(ndt \log \log n)$-bits on the size of any data structure for encoding an arbitrary graph that belongs to $\mathcal{G}_{t,d}$. We then present a $((2dt-1)n \log n + dt\log t + o(ndt \log n))$-bit data structure for $\mathcal{G}_{t,d}$ that supports navigational queries efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed $t$ and $d$, and for all $n \geq 0$ when $td^2$ is in $o(n/\log n)$ our data structure for $\mathcal{G}_{t,d}$ is succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity (denoted by $d$ and $t = 1$) and graphs of bounded interval number (denoted by $t$ and $d=1$) when $td^2$ is in $o(n/\log n)$.

Figures (5)

• Figure 1: $H \in \mathcal{G}_t, t=2$ and its 2-interval representation in (a) and (b), respectively. Observe that $H$ does not have a 2-box representation though $H$ and graph $G$, of Figure \ref{['fig:2-boxicitygraph']}, differ in only one vertex 10.
• Figure 2: $G \in \mathcal{G}_d, d=2$ and its 2-box representation in (a) and (b), respectively.
• Figure 3: (a) For $G \in \mathcal{G}^c_{2,2}$ and dependent vertex $v \in V(G)$ such that $J_v=\{(e_{1,1},e'_{1,1}),(e_{1,2},e'_{1,2}),(e_{2,1},e'_{2,1}),(e_{2,2},e'_{2,2})\}$ where $e_{1,1}=1,e'_{1,1}=2,e_{1,2}=m-1,e'_{1,2}=m,e_{2,1}=m+1,e'_{2,1}=m+2,e_{2,2}=2m,e'_{2,2}=2m$, (b) The colored neighbours of $v$ as per the basis intervals selected in (a). $S_1$ and $S_2$ contain $m$ vertices each and form an induced complete bipartite graph.
• Figure 4: Example of the data structure on $G$, represented as $2, 2$-intersection representation. Note that there also exists a $1, 1$-intersection representation of $G$.
• Figure 5: $4 \times 3$ Boolean matrix $A$ and the $3,1-$intersection representation of $G_A$.

Theorems & Definitions (36)

• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 5
• Proposition 6
• proof
• Lemma 7
• proof