Obstructions to Faster Diameter Computation: Asteroidal Sets

Full version of an IPEC'22 paperAn extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let $Ext_{\alpha}$ be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than $\alpha$ pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every $m$-edge graph in $Ext_{\alpha}$ can be computed in deterministic ${\cal O}(\alpha^3 m^{3/2})$ time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive $+1$-approximation of all vertex eccentricities in deterministic ${\cal O}(\alpha^2 m)$ time. This is in sharp contrast with general $m$-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in ${\cal O}(m^{2-\epsilon})$ time for any $\epsilon > 0$. As important special cases of our main result, we derive an ${\cal O}(m^{3/2})$-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an ${\cal O}(k^3m^{3/2})$-time algorithm for this problem on graphs of asteroidal number at most $k$. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions

Linear Time LexDFS on Cocomparability Graphs

Lexicographic depth first search (LexDFS) is a graph search protocol which has already proved to be a powerful tool on cocomparability graphs. Cocomparability graphs have been well studied by investigating their complements (comparability graphs) and their corresponding posets. Recently however LexDFS has led to a number of elegant polynomial and near linear time algorithms on cocomparability graphs when used as a preprocessing step [2, 3, 11]. The nonlinear runtime of some of these results is a consequence of complexity of this preprocessing step. We present the first linear time algorithm to compute a LexDFS cocomparability ordering, therefore answering a problem raised in [2] and helping achieve the first linear time algorithms for the minimum path cover problem, and thus the Hamilton path problem, the maximum independent set problem and the minimum clique cover for this graph family

Boundes for Boxicity of some classes of graphs

Let $box(G)$ be the boxicity of a graph $G$, $G[H_1,H_2,\ldots, H_n]$ be the $G$-generalized join graph of $n$-pairwise disjoint graphs $H_1,H_2,\ldots, H_n$, $G^d_k$ be a circular clique graph (where $k\geq 2d$) and $\Gamma(R)$ be the zero-divisor graph of a commutative ring $R$. In this paper, we prove that $\chi(G^d_k)\geq box(G^d_k)$, for all $k$ and $d$ with $k\geq 2d$. This generalizes the results proved in \cite{Aki}. Also we obtain that box(G[H_1,H_2,\ldots,H_n])\leq \mathop\sum\limits_{i=1}^nbox(H_i). As a consequence of this result, we obtain a bound for boxicity of zero-divisor graph of a finite commutative ring with unity. In particular, if $R$ is a finite commutative non-zero reduced ring with unity, then $\chi(\Gamma(R))\leq box(\Gamma(R))\leq 2^{\chi(\Gamma(R))}-2$. where $\chi(\Gamma(R))$ is the chromatic number of $\Gamma(R)$. Moreover, we show that if $N= \prod\limits_{i=1}^{a}p_i^{2n_i} \prod\limits_{j=1}^{b}q_j^{2m_j+1}$ is a composite number, where $p_i$'s and $q_j$'s are distinct prime numbers, then box(\Gamma(\mathbb{Z}_N))\leq \big(\mathop\prod\limits_{i=1}^{a}(2n_i+1)\mathop\prod\limits_{j=1}^{b}(2m_j+2)\big)-\big(\mathop\prod\limits_{i=1}^{a}(n_i+1)\mathop\prod\limits_{j=1}^{b}(m_j+1)\big)-1, where $\mathbb{Z}_N$ is the ring of integers modulo $N$. Further, we prove that, $box(\Gamma(\mathbb{Z}_N))=1$ if and only if either $N=p^n$ for some prime number $p$ and some positive integer $n\geq 2$ or $N=2p$ for some odd prime number $p$

Comparing Width Parameters on Graph Classes

We study how the relationship between non-equivalent width parameters changes once we restrict to some special graph class. As width parameters, we consider treewidth, clique-width, twin-width, mim-width, sim-width and tree-independence number, whereas as graph classes we consider $K_{t,t}$-subgraph-free graphs, line graphs and their common superclass, for $t \geq 3$, of $K_{t,t}$-free graphs. We first provide a complete comparison when restricted to $K_{t,t}$-subgraph-free graphs, showing in particular that treewidth, clique-width, mim-width, sim-width and tree-independence number are all equivalent. This extends a result of Gurski and Wanke (2000) stating that treewidth and clique-width are equivalent for the class of $K_{t,t}$-subgraph-free graphs. Next, we provide a complete comparison when restricted to line graphs, showing in particular that, on any class of line graphs, clique-width, mim-width, sim-width and tree-independence number are all equivalent, and bounded if and only if the class of root graphs has bounded treewidth. This extends a result of Gurski and Wanke (2007) stating that a class of graphs ${\cal G}$ has bounded treewidth if and only if the class of line graphs of graphs in ${\cal G}$ has bounded clique-width. We then provide an almost-complete comparison for $K_{t,t}$-free graphs, leaving one missing case. Our main result is that $K_{t,t}$-free graphs of bounded mim-width have bounded tree-independence number. This result has structural and algorithmic consequences. In particular, it proves a special case of a conjecture of Dallard, Milani\v{c} and \v{S}torgel. Finally, we consider the question of whether boundedness of a certain width parameter is preserved under graph powers. We show that the question has a positive answer for sim-width precisely in the case of odd powers.Comment: 31 pages, 4 figures, abstract shortened due to arXiv requirement