Recognizing Chordal-Bipartite Probe Graphs

A graph G is chordal-bipartite probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such that G can be extended to a chordal-bipartite graph by adding edges between non-probes. A bipartite graph is called chordal-bipartite if it contains no chordless cycle of length strictly greater than 5. Such probe/non-probe completion problems have been studied previously on other families of graphs, such as interval graphs and chordal graphs. In this paper, we give a characterization of chordal-bipartite probe graphs, in the case of a fixed given partition of the vertices into probes and nonprobes. Our results are obtained by solving first the more general case without assuming that N is a stable set, and then this can be applied to the more specific case. Our characterization uses an edge elimination ordering which also implies a polynomial time recognition algorithm for the class. This research was conducted in the context of a France-Israel Binational project, while the French team visited Haifa in March 2007

RDMA over Commodity Ethernet at Scale

ABSTRACT Over the past one and half years, we have been using RDMA over commodity Ethernet (RoCEv2) to support some of Microsoft's highly-reliable, latency-sensitive services. This paper describes the challenges we encountered during the process and the solutions we devised to address them. In order to scale RoCEv2 beyond VLAN, we have designed a DSCP-based priority flow-control (PFC) mechanism to ensure large-scale deployment. We have addressed the safety challenges brought by PFCinduced deadlock (yes, it happened!), RDMA transport livelock, and the NIC PFC pause frame storm problem. We have also built the monitoring and management systems to make sure RDMA works as expected. Our experiences show that the safety and scalability issues of running RoCEv2 at scale can all be addressed, and RDMA can replace TCP for intra data center communications and achieve low latency, low CPU overhead, and high throughput

Representations of Edge Intersection Graphs of Paths in a Tree

Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs

Graph theory, computational intelligence and thought: essays dedicated to Martin Charles Golumbic on the occasion of his 60th birthday

Published to mark the 60th birthday of Martin Charles Golumbic, whose work on algorithmic graph theory and artificial intelligence is widely celebrated, this text contains papers by graduate students, research collaborators, and computer science colleagues

Representations of Edge Intersection Graphs of Paths in a Tree

Let P be a collection of nontrivial simple paths in a tree T. The edge intersection graph of P, denoted by EP T (P), has vertex set that corresponds to the members of P, and two vertices are joined by an edge if the corresponding members of P share a common edge in T. An undirected graph G is called an edge intersection graph of paths in a tree, if G = EP T (P) for some P and T. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs. Keywords: Paths of a tree, Intersection graphs, Weakly chordal graphs, Coloring, EPT-graphs