Questions on Graph theory

Question

I am in trouble with two problems. Thank you for your positive answer.

Is the incidence matrix of a directed graph totally unimodular?
Let bipartite graph without isolated vertices. Dose the maximum cardinality of a stable set equal the minuimum cardinality of an edge cover?

dtldarek · Answer 1 · 2014-02-19T16:38:04.683

1

Hint:

For the first question, incidence matrix yes (e.g. see this pdf). Do not confuse with adjacency matrix, for example $$\det\left[\begin{array}{ccc}0&1&1\\1&0&1\\1&1&0\end{array} \right] = 2.$$
For the second question, see the König's theorem and consider the complements.

I hope this helps $\ddot\smile$

edited Feb 19 '14 at 16:38

answered Feb 19 '14 at 16:15

dtldarek

Let $S$ be a the set of all matrix such that every row is of the form $(0,...,0,1,1,...,1,0,....,0)$. Is S totally unimodular? Thank you. – user129982 Feb 20 '14 at 02:29
@user129982 Yes, it is called "consecutive-ones property", and even Wikipedia entry mentions it. – dtldarek Feb 20 '14 at 09:47
How can prove this? I tried but didnot success – user129982 Feb 24 '14 at 10:00
@user129982 I've made it a Q&A here. – dtldarek Feb 25 '14 at 10:15

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