On powers of interval graphs and their orders

It was proved by Raychaudhuri in 1987 that if a graph power $G^{k-1}$ is an interval graph, then so is the next power $G^k$. This result was extended to $m$-trapezoid graphs by Flotow in 1995. We extend the statement for interval graphs by showing that any interval representation of $G^{k-1}$ can be extended to an interval representation of $G^k$ that induces the same left endpoint and right endpoint orders. The same holds for unit interval graphs. We also show that a similar fact does not hold for trapezoid graphs.Comment: 4 pages, 1 figure. It has come to our attention that Theorem 1, the main result of this note, follows from earlier results of [G. Agnarsson, P. Damaschke and M. M. Halldorsson. Powers of geometric intersection graphs and dispersion algorithms. Discrete Applied Mathematics 132(1-3):3-16, 2003]. This version is updated accordingl

A linear-time algorithm for the strong chromatic index of Halin graphs

We show that there exists a linear-time algorithm that computes the strong chromatic index of Halin graphs.Comment: 7 page

On The Center Sets and Center Numbers of Some Graph Classes

For a set $S$ of vertices and the vertex $v$ in a connected graph $G$, $\displaystyle\max_{x \in S}d(x,v)$ is called the $S$-eccentricity of $v$ in $G$. The set of vertices with minimum $S$-eccentricity is called the $S$-center of $G$. Any set $A$ of vertices of $G$ such that $A$ is an $S$-center for some set $S$ of vertices of $G$ is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, $K_{m,n}$, $K_n-e$, wheel graphs, odd cycles and symmetric even graphs and enumerate them for many of these graph classes. We also introduce the concept of center number which is defined as the number of distinct center sets of a graph and determine the center number of some graph classes

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

Recognition of C4-free and 1/2-hyperbolic graphs

International audienceThe shortest-path metric d of a connected graph G is 1/2-hyperbolic if, and only if, it satisfies d(u,v) + d(x,y) ≤ max{d(u,x) + d(v,y),d(u,y) + d(v,x)} + 1, for every 4-tuple u,x,v,y of G. We show that the problem of deciding whether an unweighted graph is 1/2-hyperbolic is subcubic equivalent to the problem of determining whether there is a chordless cycle of length 4 in a graph. An improved algorithm is also given for both problems, taking advantage of fast rectangular matrix multiplication. In the worst case it runs in O(n^{3.26})-time

Grafos de intervalos propios y grafos arbóreos

Contenido: Introducción 1 Grafos de intervalos propios 1.1 Generalidades 1.2 Caracterizaciones 1.3 Radio y centro 1.4 Planaridad 1.5 Un problema de aplicación 2 Grafos de intervalos propios mínimos 2.1 Generalidades 2.2 Resultado Principal 2.3 Una clase clique-cerrada 2.4 Número de grafos de intervalos propios mínimos conexos 3 Grafos Arbóreo 3.1 Generalidades 3.2 Caracterizaciones 3.3 Relación con otras clases de grafos 4 Grafos de intersección 4.1 Generalidades 4.2 Una caracterización de los grafos de intersección 4.3 La aplicación dique entre ΩΣp y CΣpTesis digitalizada en SEDICI gracias a la Biblioteca del Departamento de Matemática de la Facultad de Ciencias Exactas (UNLP).Facultad de Ciencias Exacta