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Saturday 3:50–4:15 109 Oelman Hall
Results on Generalized Powers of Graphs
Yang Xiang
Kent State University
Abstract
We will present our results on ”Generalized Powers of Graphs and Their Algorithmic Use”. We will
introduce two new types of coloring of graphs, which generalize the well-known Distance-k-Coloring.
The motivation comes from the frequency assignment problem in heterogeneous multihop radio networks,
where different radio stations may have different transmission ranges. Let G = (V, E) be a
graph modelling a radio network, and assume that each vertex v of G has its own transmission radius
r(v), a nonnegative integer. We define r-coloring (r + -coloring) of G as an assignment Φ : V ↦→ 0, 1, 2, ...
of colors to vertices such that Φ(u) = Φ(v) implies dG(u, v) > r(v) + r(u) (dG(u, v) > r(v) + r(u) + 1,
respectively). The r-Coloring problem (the r + -Coloring problem) asks for a given graph G and a
radius-function r : V ↦→ N ∪0, to find an r-coloring (an r + - coloring, respectively) of G with minimum
number of colors. Using a new notion of generalized powers of graphs, we investigate the complexity
of the r-Coloring and r + -Coloring problems on several families of graphs, such as k-chordal graphs,
weakly chordal graphs, AT-free graphs, co-comparability graphs, interval graphs, circular arc graphs
and strongly chordal graphs. Most of these results were obtained in collaboration with Andreas
Brandstädt, Feodor Dragan and Chenyu Yan.
Saturday 4:15–4:40 109 Oelman Hall
Collective tree spanners of special classes of graphs
Chenyu Yan
Kent State University
Abstract
We say that a graph G = (V, E) admits a system of µ collective additive tree r-spanners if there is
a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree
T ∈ T (G) exits such that dT(x, y) ≤ dG(x, y)+r. In this talk, we give a survey of the recent results on
collective tree spanners problem. In particular, we will present the results of collective tree spanners on
chordal, AT-free related, homogeneously orderable graphs and graphs with bounded chrodality, tree
width and clique-width. Among other results we show that
• Every chordal graph admits a system of O(log n) collective additive tree 2-spanners.
• Every AT-free graphs admits a system of 2 collective additive tree 2-spanners.
• Every homogeneously orderable graph admits a system of O(logn) collective additive tree 2spanners.
• Every k-chordal graph admits a system of O(log n) collective additive tree ⌊(k +1)/3⌋-spanners.
All the above collective tree spanners can be constructed in polynomial time. This is joint work with
Derek G. Corneil and Feodor F. Dragan.
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