11,972 research outputs found
Clique-critical graphs: Maximum size and recognition
The clique graph of G, K (G), is the intersection graph of the family of cliques (maximal complete sets) of G. Clique-critical graphs were defined as those whose clique graph changes whenever a vertex is removed. We prove that if G has m edges then any clique-critical graph in K-1 (G) has at most 2m vertices, which solves a question posed by Escalante and Toft [On clique-critical graphs, J. Combin. Theory B 17 (1974) 170-182]. The proof is based on a restatement of their characterization of clique-critical graphs. Moreover, the bound is sharp. We also show that the problem of recognizing clique-critical graphs is NP-complete.Facultad de Ciencias Exacta
Small clique number graphs with three trivial critical ideals
The critical ideals of a graph are the determinantal ideals of the
generalized Laplacian matrix associated to a graph. In this article we provide
a set of minimal forbidden graphs for the set of graphs with at most three
trivial critical ideals. Then we use these forbidden graphs to characterize the
graphs with at most three trivial critical ideals and clique number equal to 2
and 3.Comment: 33 pages, 3 figure
Clique Minors in Double-critical Graphs
A connected -chromatic graph is \dfn{double-critical} if is -colorable for each edge . A long
standing conjecture of Erd\H{o}s and Lov\'asz that the complete graphs are the
only double-critical -chromatic graphs remains open for all . Given
the difficulty in settling Erd\H{o}s and Lov\'asz's conjecture and motivated by
the well-known Hadwiger's conjecture, Kawarabayashi, Pedersen and Toft proposed
a weaker conjecture that every double-critical -chromatic graph contains a
minor and verified their conjecture for . Albar and Gon\c{c}alves
recently proved that every double-critical -chromatic graph contains a
minor, and their proof is computer-assisted. In this paper we prove that every
double-critical -chromatic graph contains a minor for all . Our
proof for is shorter and computer-free.Comment: 11 pages, to appear in J. Graph Theor
Weighted network modules
The inclusion of link weights into the analysis of network properties allows
a deeper insight into the (often overlapping) modular structure of real-world
webs. We introduce a clustering algorithm (CPMw, Clique Percolation Method with
weights) for weighted networks based on the concept of percolating k-cliques
with high enough intensity. The algorithm allows overlaps between the modules.
First, we give detailed analytical and numerical results about the critical
point of weighted k-clique percolation on (weighted) Erdos-Renyi graphs. Then,
for a scientist collaboration web and a stock correlation graph we compute
three-link weight correlations and with the CPMw the weighted modules. After
reshuffling link weights in both networks and computing the same quantities for
the randomised control graphs as well, we show that groups of 3 or more strong
links prefer to cluster together in both original graphs.Comment: 19 pages, 7 figure
Between clique-width and linear clique-width of bipartite graphs
We consider hereditary classes of bipartite graphs where clique-width is bounded, but linear clique-width is not. Our goal is identifying classes that are critical with respect to linear clique-width. We discover four such classes and conjecture that this list is complete, i.e. a hereditary class of bipartite graphs of bounded clique-width that excludes a graph from each of the four critical classes has bounded linear clique-width
Homogeneous sets, clique-separators, critical graphs, and optimal -binding functions
Given a set of graphs, let be the optimal -binding function of
the class of -free graphs, that is,
In this paper, we combine the
two decomposition methods by homogeneous sets and clique-separators in order to
determine optimal -binding functions for subclasses of -free graphs
and of -free graphs. In particular, we prove the following
for each :
(i)
(ii) $\
f_{\{P_5,co-banner\}}^\star(\omega)=f^\star_{\{2K_2\}}(\omega)\in\mathcal{O}(\omega^2),\
f_{\{C_5,C_7,\ldots,banner\}}^\star(\omega)=f^\star_{\{C_5,3K_1\}}(\omega)\notin
\mathcal{O}(\omega),\ f_{\{P_5,C_4\}}^\star(\omega)=\lceil(5\omega-1)/4\rceil.G\chi(G)>\chi(G-u)u\in V(G)(P_5,banner)$-free graphs
On the Existence of Critical Clique-Helly Graphs
A graph is clique-Helly if any family of mutually intersecting cliques has non-empty intersection. Dourado, Protti and Szwarcfiter conjectured that every clique-Helly graph contains a vertex whose removal maintains it a clique-Helly graph. We will present a counterexample to this conjecture.Facultad de Ciencias Exacta
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