Bipartite powers of k-chordal graphs

Let k be an integer and k \geq 3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G^m is chordal then so is G^{m+2}. Brandst\"adt et al. in [Andreas Brandst\"adt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics, 177(1-3):9-16, 1997.] showed that if G^m is k-chordal, then so is G^{m+2}. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. Given a bipartite graph G and an odd positive integer m, we define the graph G^{[m]} to be a bipartite graph with V(G^{[m]})=V(G) and E(G^{[m]})={(u,v) | u,v \in V(G), d_G(u,v) is odd, and d_G(u,v) \leq m}. The graph G^{[m]} is called the m-th bipartite power of G. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G^{[m]}, where k, m are positive integers such that k \geq 4 and m is odd.Comment: 10 page

Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry, and in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gr\"obner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gr\"obner basis algorithms in many cases. The reason is that all computations are done on "smaller" rings, of size equal to the treewidth of the graph. In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, cryptography, sensor localization and differential equations.Comment: 40 pages, 5 figure

Low-Congestion Shortcut and Graph Parameters

Distributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of Omega~(sqrt{n} + D) rounds for many global problems, where n is the number of nodes and D is the diameter of the input graph. Since such a lower bound is derived from special "hard-core" instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts is initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. Specifically, given a specific graph class X, an f-round algorithm of constructing shortcuts of quality q for any instance in X results in O~(q + f)-round algorithms of solving several fundamental graph problems such as minimum spanning tree and minimum cut, for X. The main interest on this line is to identify the graph classes allowing the shortcuts which are efficient in the sense of breaking O~(sqrt{n}+D)-round general lower bounds. In this paper, we consider the relationship between the quality of low-congestion shortcuts and three major graph parameters, chordality, diameter, and clique-width. The main contribution of the paper is threefold: (1) We show an O(1)-round algorithm which constructs a low-congestion shortcut with quality O(kD) for any k-chordal graph, and prove that the quality and running time of this construction is nearly optimal up to polylogarithmic factors. (2) We present two algorithms, each of which constructs a low-congestion shortcut with quality O~(n^{1/4}) in O~(n^{1/4}) rounds for graphs of D=3, and that with quality O~(n^{1/3}) in O~(n^{1/3}) rounds for graphs of D=4 respectively. These results obviously deduce two MST algorithms running in O~(n^{1/4}) and O~(n^{1/3}) rounds for D=3 and 4 respectively, which almost close the long-standing complexity gap of the MST construction in small-diameter graphs originally posed by Lotker et al. [Distributed Computing 2006]. (3) We show that bounding clique-width does not help the construction of good shortcuts by presenting a network topology of clique-width six where the construction of MST is as expensive as the general case

A Dirac-type Characterization of k-chordal Graphs

Characterization of k-chordal graphs based on the existence of a "simplicial path" was shown in [Chv{\'a}tal et al. Note: Dirac-type characterizations of graphs without long chordless cycles. Discrete Mathematics, 256, 445-448, 2002]. We give a characterization of k-chordal graphs which is a generalization of the known characterization of chordal graphs due to [G. A. Dirac. On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg, 25, 71-76, 1961] that use notions of a "simplicial vertex" and a "simplicial ordering".Comment: 3 page